Modeling, Estimation, and Prototype Design for
the Cable-Actuated Bio-inspired Lightweight
Elastic Solar Sail
A THESIS
SUBMITTED TO THE FACULTY OF THE GRADUATE SCHOOL
OF THE UNIVERSITY OF MINNESOTA
BY
Nathan Raab
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF
MASTER OF SCIENCE
Prof. Ryan James Caverly
August, 2023
© Nathan Raab 2023
ALL RIGHTS RESERVED
Acknowledgements
I would like to take this opportunity to acknowledge the support I received leading
to and during the time I spent working on this thesis. This would not have been
possible without the support of everyone who helped me along the way. For that,
I am extremely grateful.
I want to thank my family for encouraging me to pursue my dreams and
providing me with the opportunities to achieve them. My mom, Sandra Thiele,
for all the tireless hours spent providing for me, answering my countless “why”
and “how” questions, and being an incredible role model. My grandpa, Randal
Gascoigne, for inspiring my passion for electrical engineering and robotics. And
my grandma, Sharon Gascoigne, for teaching me everything else.
I also want to thank my advisor, Professor Ryan Caverly, for his guidance
and support during my time in the Aerospace, Robotics, Dynamics, and Control
(ARDC) Lab. His valuable insight and knowledge were key to my success. I am
grateful for the opportunity he provided to work on this project and the invaluable
learning experiences I gained during my time working on this thesis.
Finally, I would like to acknowledge the support I received from my friends
in the ARDC lab during this project. Whether it was providing ideas or talking
through problems, everyone was more than willing to help me overcome the chal-
lenges I faced. I would also like to extend special thanks to all of the team members
who helped with the CABLESSail project. Thanks to the graduate team, Kee-
gan Bunker and Soojeong Lee. Thanks to the senior design team, Garvin Saner,
Zixin Chen, Tyler Douvier, Richard J. Lyman, Owen Sorby, Benjamin Sorge,
i
Ebise Teshale, and Benjamin Toriseva, who helped lay the groundwork for much
of the solar sail theory and preliminary analysis. And thanks to Michael Dallalah,
Michael States, and Niko Sexton, who helped design and test the CABLESSail
prototypes.
ii
Dedication
To my family, especially my mother and grandparents.
iii
Abstract
Traditional spacecraft that utilize rocket engines for propulsion have a fundamen-
tal drawback in that they require the use of rocket fuel. As a result, there is a
need to develop spacecraft with alternative means of propulsion that can extend
the effective mission range without relying on heavy, expensive rocket fuel. One
of the most promising alternative means of propulsion is the solar sail. These sails
utilize the reflection of solar radiation pressure and the corresponding transfer of
momentum to achieve propulsion. However, one of the most significant challenges
pertaining to solar sail design is the means by which stabilization and attitude
control are achieved. To address these challenges, this thesis presents the concept
of the Cable-Actuated Bio-inspired Lightweight Elastic Solar Sail (CABLESSail).
This novel solar sail concept uses cable-actuated flexible beams to achieve sta-
bilization and attitude control by means of sail shape modulation. Analysis is
shown that confirms the feasibility of the CABLESSail model by demonstrating
that NASA mission attitude control requirements for both Solar Cruiser and So-
lar Polar Imager can be achieved through cable actuation. A method to predict
the end-effector pose given the cable lengths or cable tension is discussed, with in-
tended application in a control algorithm to control the end-effector pose. A proto-
type cable-driven continuum manipulator is designed for use as the cable-actuated
flexible beam. The cable-driven continuum manipulator is modeled in simulation,
and the simulation results of the beam deflection, induced cable torque, and the
change in motor position are compared to experimental results.
iv
Contents
Acknowledgements i
Dedication iii
Abstract iv
List of Tables viii
List of Figures xi
1 Introduction 1
1.1 Motivation for the Cable-Actuated Bio-inspired Lightweight Elastic
Solar Sail . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Thesis Outline and Contributions . . . . . . . . . . . . . . . . . . 5
2 Theory of the Cable-Actuated Bio-inspired Lightweight Elastic
Solar Sail 7
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 CABLESSail Concept and Model . . . . . . . . . . . . . . . . . . 8
2.3 CDCM Concept and Model . . . . . . . . . . . . . . . . . . . . . 10
2.4 Technical Requirements . . . . . . . . . . . . . . . . . . . . . . . 12
2.5 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.5.1 Pitch/Yaw Torque Simulation Results . . . . . . . . . . . . 21
2.5.2 Roll Torque Simulation Results . . . . . . . . . . . . . . . 28
v
2.5.3 Propulsion Force Losses Discussion and Simulation Results 33
2.6 Closing Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3 Deriving the Forward Kinematics of the Cable-Driven Continuum
Manipulator 38
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.2 Polynomial Fitting for the Forward Kinematics of the CDCM . . 39
3.2.1 Polynomial Fitting for the Forward Kinematics of the
CDCM using Uniformly Spaced Data . . . . . . . . . . . . 40
3.2.2 Polynomial Fitting for the Forward Kinematics of the
CDCM using Chebyshev Nodes . . . . . . . . . . . . . . . 43
3.2.3 Polynomial Fitting Discussion . . . . . . . . . . . . . . . . 47
3.3 Assumed Modes Method for the Forward Kinematics of the CDCM 50
3.4 Neural Network Method for the Forward Kinematics of the CDCM 55
3.5 Closing Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4 Hardware Design of the Cable-Actuated Bio-inspired Lightweight
Elastic Solar Sail 65
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.2 Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.2.1 Iteration I . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.2.2 Iteration II . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.2.3 Iteration III . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.3 Hardware Selection . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
4.5 Closing Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
5 Modeling the Dynamics of the Cable-Driven Continuum Manip-
ulator Hardware 88
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
5.2 Permanent Magnet Synchronous Motor Concepts . . . . . . . . . 89
vi
5.3 Modeling the ODrive D6374 . . . . . . . . . . . . . . . . . . . . . 96
5.4 Controller Design for the ODrive D6374 . . . . . . . . . . . . . . 102
5.5 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . 107
5.6 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . 114
5.7 Closing Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
6 Conclusion 128
References 130
Appendix A. Solar Sail Beam Deformation for the Collapsible
Tubular Mast Beam 136
Appendix B. Additional Cable-Driven Continuum Manipulator
Figures 139
B.1 Iteration I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
B.2 Iteration III . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
Appendix C. Prototype Cable-Driven Continuum Manipulator
Parts List 145
vii
List of Tables
2.1 NASA requirements for attitude control of Solar Cruiser and Solar
Polar Imager. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2 Alternative NASA requirements for attitude control of Solar
Cruiser and Solar Polar Imager. . . . . . . . . . . . . . . . . . . . 14
2.3 Combined NASA requirements for attitude control of Solar Cruiser
and Solar Polar Imager. . . . . . . . . . . . . . . . . . . . . . . . 14
2.4 Material properties for Hexcel® IM7/977-2. . . . . . . . . . . . . 15
2.5 Area moment of inertia properties of the TRAC boom cross-section
as modeled in Solidworks . . . . . . . . . . . . . . . . . . . . . . . 16
2.6 Material properties for the NEA Scout Sail. . . . . . . . . . . . . 20
2.7 Cable tension, cable length displacement, and boom displacement
required for attitude control to achieve double the NASA pitch/yaw
torque requirements . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.8 Cable tension, cable length displacement, and boom displacement
required for stabilization control to achieve double the NASA
pitch/yaw torque requirements . . . . . . . . . . . . . . . . . . . . 28
2.9 Boom twist required to achieve double the NASA roll torque
requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.10 Boom twist required to achieve double the NASA roll torque
requirements using the trapezoidal method . . . . . . . . . . . . . 33
3.1 R2 values of the polynomial fit function applied to uniformly spaced
data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
viii
3.2 RMSE values of the polynomial fit function generated from
uniformly spaced data applied to randomly generated data. . . . . 42
3.3 Functions representing the equilibrium solution for the CDCM pose
vs. cable length. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.4 R2 values of the polynomial fit function applied to data generated
from the Chebyshev node distribution. . . . . . . . . . . . . . . . 45
3.5 RMSE values of the polynomial fit function generated from the
Chebyshev node distribution applied to randomly generated data. 46
3.6 Functions representing the equilibrium solution for the CDCM pose
vs. cable length using the Chebyshev node distribution. . . . . . . 47
3.7 RMSE values of the polynomial fit function generated from the
Chebyshev random distribution. . . . . . . . . . . . . . . . . . . . 49
3.8 Parameters of the neural network gradient equations . . . . . . . 58
3.9 Average RMSE of the pose with respect to the number of epochs. 60
3.10 Average RMSE of the pose with respect to the number of layers. . 60
3.11 Average RMSE of the pose with respect to the number of layers. . 60
3.12 Average RMSE of the pose with respect to the method of data
generation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.13 Average RMSE of the pose with respect to the number of layers,
considering only epochs 101, 251, and 501. . . . . . . . . . . . . . 61
3.14 Average RMSE of the pose with respect to the number of layers,
considering only epochs 101, 251, and 501. . . . . . . . . . . . . . 61
3.15 Average RMSE of the pose with respect to the method of data
generation, considering only epochs 101, 251, and 501. . . . . . . . 62
4.1 Area moment of inertia properties of the tape measure TRAC boom
cross-section. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.2 Material properties for polycarbonate . . . . . . . . . . . . . . . . 79
4.3 Material properties for a C1095 steel alloy . . . . . . . . . . . . . 80
4.4 Functions representing the equilibrium solution for the polycarbon-
ate beam displacement and motor torque vs. rotations. . . . . . . 81
ix
4.5 Functions representing the equilibrium solution for the tape
measure TRAC boom displacement and motor torque vs. rotations. 81
4.6 Simulation results for the 3 ft polycarbonate beam using the PRB
model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
4.7 Simulation results for the 2.5 m tape measure TRAC boom using
the PRB model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
4.8 Experimental equilibrium results for the 3 ft polycarbonate beam. 85
4.9 Simulation results for the 3 ft polycarbonate beam using the PRB
model and the experimental torques for each set point. . . . . . . 85
5.1 Parameters of the PMSM equations. . . . . . . . . . . . . . . . . 95
5.2 Motor properties for the ODrive D6374. . . . . . . . . . . . . . . 97
5.3 Inferred and measured motor properties for the ODrive D6374. . . 98
5.4 Experimental parameters and derived coefficients for the static
friction evaluation using the ODrive D6374. . . . . . . . . . . . . 99
5.5 Experimental parameters for the kinetic friction evaluation using
the ODrive D6374. . . . . . . . . . . . . . . . . . . . . . . . . . . 101
5.6 Kinetic friction coefficients for the ODrive D6374. . . . . . . . . . 101
5.7 Parameters of the ODrive D6374 controller . . . . . . . . . . . . . 103
5.8 Simulated and experimental deflection of the end-effector with the
polycarbonate beam. . . . . . . . . . . . . . . . . . . . . . . . . . 119
5.9 Motor position corresponding to the experimental end-effector de-
flection and the resulting position ratio between the corresponding
motor position and the motor set point. . . . . . . . . . . . . . . . 120
C.1 Parts list for all components required for construction of the third
iteration of the CDCM. . . . . . . . . . . . . . . . . . . . . . . . . 150
x
List of Figures
2.1 Schematic of the CABLESSail model. . . . . . . . . . . . . . . . . 8
2.2 Schematic of the CABLESSail concept of operation. . . . . . . . . 9
2.3 Example CDCM schematic with twelve cable elements. . . . . . . 10
2.4 CDCM model with two direct cables and two helical cables. . . . 11
2.5 Cross-sectional models of three boom designs used in NASA solar
sail projects. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.6 Dimensions of the TRAC boom used in NASA solar sail projects. 16
2.7 Model of the TRAC boom cross-section in Solidworks . . . . . . . 17
2.8 Lateral boom deflection vs. cable tension for the chosen disk radii
of 2 cm, 4 cm, 6 cm, 8 cm, and 10 cm. . . . . . . . . . . . . . . . 20
2.9 Pitch/Yaw torque generated from lateral boom deformation for
Solar Cruiser with a 2.0 cm cable radius and 85 N of maximum
tension. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.10 Pitch/Yaw torque generated from lateral boom deformation for
Solar Polar Imager with a 2.0 cm cable radius and 67 N of maximum
tension. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.11 Maximum beam deformation required for attitude control. . . . . 26
2.12 Pitch/Yaw torque generated from lateral boom deformation for
Solar Cruiser with a 2.0 cm cable radius and 34 N of maximum
tension. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
xi
2.13 Pitch/Yaw torque generated from lateral boom deformation for
Solar Polar Imager with a 2.0 cm cable radius and 17 N of maximum
tension. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.14 Maximum beam deformation required for stabilization control. . . 28
2.15 Roll torque generated from boom twist for Solar Cruiser with a
15 cm spreader length. . . . . . . . . . . . . . . . . . . . . . . . . 30
2.16 Roll torque generated from boom twist for Solar Polar Imager with
a 60 cm spreader length. . . . . . . . . . . . . . . . . . . . . . . . 30
2.17 Roll torque generated from boom twist for Solar Cruiser with a
15 cm spreader length using the finite element trapezoid method. 32
2.18 Roll torque generated from boom twist for Solar Polar Imager with
a 60 cm spreader length using the finite element trapezoid method. 32
2.19 Propulsion (normal) force with respect to lateral cable actuation
for Solar Cruiser. . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.20 Propulsion (normal) force with respect to lateral cable actuation
for Solar Polar Imager. . . . . . . . . . . . . . . . . . . . . . . . . 35
2.21 Propulsion (normal) force with respect to torsional cable actuation
for Solar Cruiser using the trapezoid method. . . . . . . . . . . . 36
2.22 Propulsion (normal) force with respect to torsional cable actuation
for Solar Polar Imager using the trapezoid method. . . . . . . . . 36
3.1 Illustration of Runge’s Phenomenon, and the advantage of Cheby-
shev interpolation (CIP) over other interpolation methods. In this
case, Lagrange interpolation (LIP). . . . . . . . . . . . . . . . . . 44
3.2 Cantilever beam simulation plotted over time along with its
equilibrium position. . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.3 Cantilever beam simulation plotted over time along with its
bounding envelope. . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.4 Illustration of a standard perceptron with an arbitrary activation
function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.5 Illustration of a standard neural network with a single output. . . 56
xii
4.1 Isometric model of the first iteration of the CDCM base. . . . . . 67
4.2 Isometric model of the first iteration of the CDCM beam elements. 67
4.3 Isometric model of the second iteration of the CDCM beam elements. 68
4.4 Isometric model of the third iteration of the CDCM base. . . . . . 69
4.5 Isometric model of the polycarbonate beam insert for the third
iteration of the CDCM. . . . . . . . . . . . . . . . . . . . . . . . . 70
4.6 Isometric model of the tape measure beam insert for the third
iteration of the CDCM. . . . . . . . . . . . . . . . . . . . . . . . . 70
4.7 Isometric model of the third iteration of the CDCM beam elements. 71
4.8 Isometric model of the beam elements used for the tape measure
TRAC boom. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.9 Model of the tape measure TRAC boom cross-section. . . . . . . 73
4.10 Hardware selection for the CDCM. . . . . . . . . . . . . . . . . . 75
4.11 Arduino Giga R1 used for CDCM control . . . . . . . . . . . . . . 76
4.12 Polycarbonate beam for the CDCM. . . . . . . . . . . . . . . . . 76
4.13 Tape measure TRAC boom for the CDCM. . . . . . . . . . . . . . 77
4.14 Fully assembled third iteration of the CDCM. . . . . . . . . . . . 77
4.15 (a) The SureStep regeneration clamp and (b) the Mean Well LRS-
600-24 power supply. . . . . . . . . . . . . . . . . . . . . . . . . . 78
4.16 Experimental results for the CDCM with the polycarbonate beam. 83
5.1 Illustration of the Clarke and Park transformation from a three-
phase static reference frame to a two-parameter rotating reference
frame. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
5.2 Equivalent q-axis circuit diagram of a three-phase PMSM after
applying the DQZ transformation, assuming zero d-axis current. . 93
5.3 Interior of the KDE Direct KDE2315XF-885, a hobby-grade
outrunner motor similar to the ODrive D6374. . . . . . . . . . . . 96
5.4 Circuit diagram of a three-phase PMSM, such as the ODrive D6374. 97
5.5 Total friction vs. motor velocity for the ODrive D6374. . . . . . . 101
xiii
5.6 Cascaded position, velocity, and current controller implemented by
ODrive. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
5.7 Simulation results for the CDCM with the TRAC boom assuming
constant voltage. . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
5.8 Power requirement simulation results for the CDCM with the
TRAC boom assuming constant voltage. . . . . . . . . . . . . . . 109
5.9 Simulation results for the CDCM with the TRAC boom using
torque control. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
5.10 Power requirement simulation results for the CDCM with the
TRAC boom using torque control. . . . . . . . . . . . . . . . . . . 110
5.11 Simulation results for the CDCM with the TRAC boom using
position control. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
5.12 Power requirement simulation results for the CDCM with the
TRAC boom using position control. . . . . . . . . . . . . . . . . . 114
5.13 Simulation and experimental motor position results for the CDCM
with the polycarbonate beam using position control. . . . . . . . . 116
5.14 Simulation and experimental motor torque results for the CDCM
with the polycarbonate beam using position control. . . . . . . . . 117
5.15 Simulation and experimental end-effector deflection results for the
CDCM with the polycarbonate beam using position control. . . . 118
5.16 Modified simulation and experimental motor position results for
the CDCM with the polycarbonate beam using position control. . 121
5.17 Modified simulation and experimental motor torque results for the
CDCM with the polycarbonate beam using position control. . . . 122
5.18 Modified simulation and experimental end-effector deflection re-
sults for the CDCM with the polycarbonate beam using position
control. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
5.19 Mechanical power requirement simulation and experimental results
for the CDCM with the polycarbonate beam. . . . . . . . . . . . 125
xiv
5.20 Electrical power losses simulation and experimental results for the
CDCM with the polycarbonate beam. . . . . . . . . . . . . . . . . 126
A.1 Pitch/Yaw torque generated from lateral boom deformation for
Solar Cruiser with a 2 cm cable radius and 8.2 N of maximum
tension. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
A.2 Pitch/Yaw torque generated from lateral boom deformation for
Solar Polar Imager with a 2 cm cable radius and 6.45 N of maximum
tension. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
B.1 Engineering Diagram of the first iteration of the CDCM base. . . 140
B.2 Isometric model of the first iteration of the CDCM base with the
motor enclosure removed. . . . . . . . . . . . . . . . . . . . . . . 141
B.3 Engineering Diagram of the third iteration of the CDCM base. . . 142
B.4 Isometric model of the third iteration of the CDCM base with the
motor enclosures removed. . . . . . . . . . . . . . . . . . . . . . . 143
B.5 Isometric model of the interior of the third iteration of the CDCM
base. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
B.6 Isometric model of the tape measure TRAC boom with attached
beam elements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
xv
Chapter 1
Introduction
1.1 Motivation for the Cable-Actuated
Bio-inspired Lightweight Elastic Solar Sail
Since the advent of space flight, the traditional means of propulsion in space has
been through the use of rocket engines. However, traditional rocket engines have
one major drawback: they require a significant amount of fuel. In particular,
a typical planetary-exploration spacecraft carries up to 25 percent of its launch
weight as rocket fuel [1]. This is a significant limitation to the mission capabilities
of a traditional rocket-fueled spacecraft. As a result, one of the main focuses in
spaceflight research is on ways to increase the efficiency of space travel. One of
the most promising areas of research is the concept of solar sails.
Solar sails are structures that use solar radiation pressure to serve as the means
of propulsion for a spacecraft [2]. Like a classical sailing vessel that uses wind to
propel itself forward, a solar sail relies on photons hitting its sail for propulsion.
This has advantages of being both extremely efficient and lightweight compared
to traditional spacecraft since the solar sail uses no fuel. These fuel savings could
theoretically allow for a host of new mission possibilities. For example, the ef-
ficiency of solar sails can be utilized to achieve orbits outside the ecliptic plane,
1
2non-keplerian orbits, and interstellar travel [3–6].
The first solar sails have just recently begun to take flight. In 2005, the first
solar sail mission ever attempted was launched, named Cosmos 1 [7]. However,
the spacecraft never reached its intended orbit, and the mission was a failure.
The first successful solar sail mission was IKAROS, launched by the Japanese
Aerospace Exploration Agency in 2010 [7]. Tension in the sail was maintained
using centripetal acceleration. Attitude control was achieved using 80 liquid crys-
tal display (LCD) panels whose reflectivity could be controlled. At 196 square
meters, this is also the largest solar sail successfully tested to date [8]. Since
the launch of IKAROS, the majority of solar sails have been CubeSats - small,
lightweight, cube-shaped spacecraft. These include the NanoSail-D2, LightSail 1
and LightSail 2, and Near-Earth Asteroid (NEA) Scout [7]. These solar sails had
sail areas of 10 m2, 32 m2, 32 m2, and 86 m2, respectively. However, as discussed
in [7], transitioning from proof of concept to advanced mission concepts will re-
quire significantly larger solar sail designs. This is because the force induced by
solar radiation pressure is very small [9]; at 1 AU, the force induced by solar ra-
diation pressure is only 9.126 µN/m2. Some of these conceptual designs include
SunJammer, which would have a sail area of over 1200 m2, Solar Cruiser, which
would have a sail area of 1653 m2, and Solar Polar Imager, which would have a sail
area of over 7000 m2 [10–12]. However, as solar sails grow in size, new design chal-
lenges emerge. In particular, the inertia of the spacecraft increases significantly,
and the structural rigidity of the solar sail beams becomes a concern.
Some of the most significant challenges related to solar sail development are
the method of attitude control and its overall structural integrity [7]. In many
ways, these challenges are intertwined with each other and with the size of the sail.
As the solar sail increases in size, it will tend to increase in both mass and inertia.
Correspondingly, the attitude control problem becomes more challenging, as more
control authority is needed to move a more massive sail. Yet, relying on a heavy
mass attitude control method would limit the effectiveness of the sail’s propulsion.
Similarly, as the sail becomes physically larger, the material properties of the sail
3must improve in order to maintain structural integrity.
Many solutions have been proposed to provide attitude control for solar sails.
The most straightforward approach simply uses conventional thrusters attached
to the spacecraft to provide attitude control [13]. However, this limits the advan-
tage of the sail’s propellantless design. Another method uses reaction wheels for
attitude control. This is a very common approach utilized by many spacecraft.
However, as the size and inertia of the solar sail increase, the control authority of
the reaction wheels must similarly increase. In general, this means the mass of the
reaction wheels must increase, which increases the mass of the overall spacecraft.
For solar sails, where mass is a significant concern, larger reaction wheels are not
always a feasible solution. A somewhat similar approach uses active mass control
for attitude control [13]. Several different methods of active mass control have
been proposed for solar sail attitude control. The first is the “two-axis gimbal”
method. In this design, the spacecraft uses control masses located near the center
of mass and shifts the center of mass with respect to the center of pressure to
induce a torque [13]. However, this method is strongly limited by the uncertainty
in the center of mass and center of pressure. This method also only achieves two-
dimensional attitude control. Thus, this method would have to be combined with
another form of attitude control to achieve complete control authority. The second
is the “translating masses” method. In this design, masses are mounted along the
booms of the sail and can be translated to augment the center of mass. Similar to
the two-axis gimbal method, this method only achieves two-dimensional attitude
control. One method unique to solar sails is to attach tip vanes to the end of the
sail booms [13]. Similar to airplane control surfaces, this method uses reflective
surfaces that rotate in one or two axes in order to achieve differential momentum
transfer and thus a net torque. However, these vanes can be quite complex and
large in mass, which can increase the mass and inertia of the sail. Since they also
provide control authority using solar radiation pressure, the control authority of
the sail depends on the solar irradiance observed by the sail [14]. A relatively
4unique approach involves the use of reflectivity control to achieve control author-
ity. The most notable example of this method is the reflectivity control devices
used by IKAROS [7]. By controlling the reflectivity of 80 LCDs mounted on the
edge of the sail, differential reflectivity can be induced and the orientation of the
sail can be controlled. However, this necessitates the addition of LCD panels along
with electrical wiring and a power source to control the reflectivity of the panels,
complicating the design of the spacecraft [15]. Further, the control authority of
the reflectivity control devices can degrade over time [15]. Finally, some literature
has proposed to implement three-axis control by precisely deforming the solar
sail [16]. In [16, 17], the boom itself is controlled by attachment to a gimbaled
control boom. But for very large sails, the gimbal needs to be strong enough to
stabilize a very long boom. In [17], the idea of implementing control authority
based on the deformation of the flexible sail film is introduced. However, this
paper does not describe the method by which the sail film is controlled.
As the size of the solar sail spacecraft increases, the structural integrity of
the booms must be taken into consideration. In general, solar sails can either
utilize booms or rely on centripetal acceleration through constant spin motion to
maintain sail tension [7]. A spacecraft that relies on constant rotation to maintain
stability has several clear drawbacks. In particular, the attitude control becomes
significantly more complex, and the lack of support structure means the sail is
prone to oscillations. However, a sail that uses booms has several disadvantages
as well. The storage and deployment of the booms is non-trivial, and results in
booms that need to be stored compactly. This limits the structural rigidity of the
booms since they must be sufficiently flexible as required for compact storage. As
a result, large sails utilizing the boom structure may still need to undergo rotation
to prevent the booms from buckling.
This thesis proposes a novel approach to solar sail design based on the princi-
ples of the cable-driven continuum manipulator (CDCM). In this design, each of
the sail booms is flexible. The flexible boom has several backbone structures, or
disks, with cables that are threaded through these discs in a particular manner
5to achieve control authority. In particular, with two directly threaded and two
helically threaded cables in each of the four booms of a solar sail, full three-axis
control authority can be achieved. Furthermore, by maintaining positive tension
in the cables, structural integrity of the sail can be ensured.
1.2 Thesis Outline and Contributions
This thesis develops the concept of the Cable-Actuated Bio-inspired Lightweight
Elastic Solar Sail (CABLESSail), a novel solar sail design approach that uses
cables to actuate the sail beams in order to provide attitude control. The design
criteria for the solar sail are given by NASA and interpreted in the context of the
CABLESSail design. This provides design criteria for the CDCM, which is the
fundamental robot model that controls the deflection of the CABLESSail beams.
This design is implemented in hardware and compared with the results predicted
by simulation. The structure of this thesis and the major contributions of each
chapter are outlined in the following paragraphs.
Chapter 2 provides a detailed background of solar sail concepts in order to in-
terpret the NASA mission requirements in the context of the CABLESSail model.
In particular, the CABLESSail design is analyzed in the context of both Solar
Cruiser and Solar Polar Imager. The sail is modeled using a high-fidelity solar
radiation pressure force equation that considers the particular sail material prop-
erties. The boom properties of the TRAC boom are modeled in simulation to
determine the magnitude of deformation for a particular cable tension. Finally,
the resulting roll, pitch, and yaw attitude control torques are determined for a
sweep of beam deformations in order to determine the particular beam deforma-
tion that meets the NASA requirements.
Chapter 3 discusses the estimation of the end-effector pose of the CDCM given
the measurements available to the robot. Three different methods are discussed
and compared. The first method is the polynomial line of best fit using inter-
polation of the static equilibrium data. The second method solves the nonlinear
6equilibrium solution of a cantilever beam model of the CDCM. The third method
implements a neural network to predict the end-effector pose.
Chapter 4 steps through the hardware design and build process for each of the
three iterations of the prototype CDCM. The control methods, hardware selection,
and structural design rationale are provided. Experimental data is collected and
compared with the simulation models developed in Chapters 2 and 3.
Chapter 5 derives a dynamic model of the permanent magnet synchronous
motor used for the actuation of the prototype CDCM. The theory and governing
equations of the motor are provided. The motor controller is implemented using
the field-oriented control technique. This model is compared in simulation to the
experimental results found in Chapter 4.
Chapter 2
Theory of the Cable-Actuated
Bio-inspired Lightweight Elastic
Solar Sail
2.1 Introduction
The concept of the CABLESSail is built upon the concept of the cable-driven
continuum manipulator (CDCM). In order to develop a scale prototype of the
CABLESSail concept, the CDCM must first be designed to provide actuation of
the booms. In addition, this design must integrate with existing NASA designs
and mission criteria for solar sails. Building on the work of a senior design team
at the University of Minnesota, this chapter details the modeling of the CDCM
using the NASA-developed Triangular Rollable And Collapsible (TRAC) boom
design. This chapter provides a quantitative analysis of the forces required to gen-
erate the required boom deflection in order to meet NASA mission requirements.
In addition, this chapter quantifies the propulsive force losses associated with a
deflected sail.
7
82.2 CABLESSail Concept and Model
The CABLESSail model is a solar sailing vessel that integrates cable-controlled
flexible booms in order to precisely deform the sail and provide attitude and
stability control [18]. The cables are routed along the flexible booms through
discrete beam elements and connect to an actuator at the base of the boom, such
as a motor and winch. As the torque of the motor is controlled, the cable tension
is modified and induces a reaction force that pulls on the boom. The magnitude
of cable tension corresponds precisely to the magnitude of beam deflection. In
turn, the magnitude of boom deflection induces a solar radiation pressure (SRP)
disparity across the sail, which induces torque. This torque induces a roll, pitch,
or yaw motion of the sail, depending on cable actuation. A schematic of the
CABLESSail concept is shown in Fig. 2.1.
b
−→
3
b
−→
1
b
−→
2Sail
Flexible Boom
Spreader
Bus
b
−→
1
b
−→
2
Flexible Boom
Rigid Boom Location
T2
T3
T4
T2 and T3 Twist the Boom/Sail
T1 and T4 Bend the Boom/Sail
Winches in Bus Actuate T1, T2, T3, T4:
T1
Spreaders
b
−→
2
b
−→
1
Torque About b
−→
3 (Yaw)
Bending InducesExample:
Figure 2.1: Schematic of the CABLESSail model [18].
Each cable induces a single-axis beam deformation. The axis of deformation
depends on the particular cable routing along the beam, either direct or helical.
Direct cable routing provides a lateral deformation in the direction of the cable
with respect to the axis of the boom, while helical cable routing provides a tor-
sional deformation about the axis of the boom. In order to achieve the maximum
3-DoF of control authority, a total of eight cables are needed - four helically routed
cables, and four direct routed cables. However, in the CABLESSail concept, only
9one axis of lateral deformation is needed. In certain cable actuation designs, the
number of cables can be further reduced if torsional and lateral deformation are
only needed in a single direction. In these cases, the total number of cables can
be reduced to six or three, respectively. This design is inspired by the concept of
the CDCM, which is discussed in the following section.
The concept by which the beam deformation induces a net torque across the
sail is illustrated in Fig. 2.2. Illustrated is an ideal solar sail with nominal deflec-
tion, along with a qualitative representation of the induced SRP force disparity.
In each case, the photons reflecting off the surface transfer their momentum to
the spacecraft. This can be quantified as the magnitude of the SRP force. As
the sail area is deflected, the angle of the photon reflection is modified. As the
magnitude of the deflection angle is increased with respect to the sun, the normal
component of the photon reflection is reduced and the translational component is
increased. As a result, the induced SRP force is no longer constant across the sail.
The magnitude of induced torque is therefore determined based on the differential
SRP force acting on the sail.
b
−→
2
b
−→
1
direction induces torque about b
−→
3 (yaw)
Bending of one or both booms in b
−→
1
b
−→
3
b
−→
1
b
−→
2
b
−→
2
b
−→
1
δ1 δ2
δ2
2-Boom Bending:
1-Boom Bending:
(a)
b
−→
2
b
−→
1
axes induces torque about b
−→
2 (roll)
Twisting all booms about their longitudinal
b
−→
3
b
−→
1
b
−→
2
δ3
(b)
Figure 2.2: Schematic of the CABLESSail concept of operation. Shown are the
(a) lateral boom deformation and (b) torsional boom deformation.
The CABLESSail concept is designed for integration with existing NASA mod-
els for both Solar Cruiser and Solar Polar Imager. NASA requirements are used to
develop the design criteria for the CABLESSail concept, including the magnitude
of deflection and actuator control authority.
10
2.3 CDCM Concept and Model
The CDCM is a modified suspended cable-driven parallel robot (CDPR) that is
constrained by a flexible beam in place of, or in addition to, the force of gravity. In
addition, the CDCM introduces a backbone structure that threads the cables from
the base of the manipulator to the end-effector. As a result, the deformation of
the beam is approximately constant. The cables are controlled by a load, typically
a winch attached to a motor located behind the base of the CDCM. An example
CDCM schematic is shown in Fig. 2.3.
Figure 2.3: Example CDCM schematic with twelve cable elements [19].
In this thesis, the CDCM considered is a 4-actuator, 2-DoF robot. Hence,
the CDCM is over-constrained. Each of the degrees of freedom is controlled by
two parallel motors. These motors apply torque to control the tension in their
respective cables. Two of these cables are threaded directly to provide lateral de-
formation and two cables are threaded helically to provide torsional deformation.
This is illustrated in Fig. 2.4.
11
uˆ1
x1
x3
x2
uˆ2uˆ3 uˆ4
Fn
Base
DiscsDirect Cables
Helical Cables
Flexible Boom
Figure 2.4: CDCM model with two direct cables and two helical cables. Note that
the number of discs can be modified and the dimensions are not to scale.
The pose of the CDCM is described by a 7-state vector consisting of the
position and the quaternion of the end-effector of the beam. Explicitly, the state
vector is
z =
[
x1 x2 x3 qw qx qy qz
]⊤
, (2.1)
where the quaternion coefficients are given by
qw + qxi+ qyj+ qzk. (2.2)
The physical model of a CDCM is both complex and nonlinear. Significant
work has been conducted on the modeling of CDCMs at the University of Min-
nesota. Work from Vinh Nguyen in the Aerospace, Robotics, Dynamics, and
Control (ARDC) Lab has modeled the dynamics of a CDCM given some input
parameters and the physical parameters of the CDCM. With this, it is shown
that the CDCM can achieve both lateral and torsional deformation, as required
to provide full 3-axis control of a solar sail.
12
2.4 Technical Requirements
Preliminary work from a senior design team at the University of Minnesota has
benchmarked the magnitude of deformation required to achieve NASA mission
criteria for Solar Cruiser and Solar Polar Imager at particular solar incidence an-
gles (SIAs). Their analysis utilized a CDCM model developed by the University
of Toronto Mississauga [20]. This code allows for the particular CDCM beam
properties to be modeled in simulation in order to predict the precise beam de-
formation given applied cable tensions. In addition, the gravitational acceleration
can be modified to reflect Earth’s gravity or to simulate operation in zero grav-
ity. The gravitational acceleration parameter was set to zero since the intended
operation of the CDCM is in space.
This code had five different methods to model a CDCM, each with trade-
offs between accuracy and computation time. For their analysis of lateral beam
deformation, the senior design team selected the constant curvature subsegment
(CCsub) method. This method assumes that each subsegment between beam el-
ements retains a constant curvature. Compared to other methods, this method
is more accurate than the constant curvature (CC) approach, but faster compu-
tationally (but less accurate) than the variable curvature and pseudo-rigid body
approaches. For this analysis, the loss of accuracy was deemed acceptable in order
to minimize computation time.
The simulation model developed in this chapter seeks to improve upon the
work of the senior design team, which was limited in several respects. First, their
analysis approximated the boom material properties using a generic carbon-fiber
beam. Second, the analysis assumed only one boom deformation at a time. Third,
the analysis only investigated SIA defined with respect to the axis of deformation,
but did not investigate SIA for different clock angles. Fourth, their torsional
deformation analysis assumed the sail connects only at the tip of the boom, and
thus the sail deformation is approximated as a triangle. This triangle is defined
by the vector from the connection point at the satellite to the twisted end-effector
13
of the boom. This is not accurate with respect to the deformation caused by
helically threaded cables, particularly for a sail with several beam elements along
the boom. Finally, the analysis used an approximation for reflectivity rather than
considering the material properties of the sail. Building off of the model developed
in [20], these aspects of the analysis are extended to more accurately model the
existing technology in use by NASA.
The mission requirements for both Solar Cruiser and Solar Polar Imager are
outlined in [21] and are provided in Table 2.1. However, different documentation
Case Out-of-Plane (Roll)
Torque (Nm)
In-Plane (Pitch/Yaw)
Torque (Nm)
Solar Cruiser
(1650 m2) 0° SIA
4.7×10-6 5.5×10-4
Solar Cruiser
(1650 m2) 35° SIA
4.9×10-5 2.4×10-3
Solar Polar Imager
(7000 m2) 0° SIA
7.7×10-5 4.6×10-3
Solar Polar Imager
(7000 m2) 17° SIA
6.4×10-4 4.7×10-2
Table 2.1: NASA requirements for attitude control of Solar Cruiser and Solar
Polar Imager [21].
provides slightly different mission criteria for both Solar Cruiser and Solar Polar
Imager [22]. In particular, the in-plane torque requirements for Solar Cruiser at 0°
SIA and the SIA requirements for Solar Polar Imager are modified. These modified
criteria are provided in Table 2.2. Noting the discrepancy between these two re-
quirements, the more extreme criterion is considered for each of the requirements.
These combined requirements are outlined in Table 2.3.
NASA outlines three flexible boom designs for use in its solar sail projects [23].
These are named the Storable Tubular Extendable Member (STEM), the Collapsi-
ble Tubular Mast (CTM), and the TRAC boom. The cross-sectional models of
these designs are illustrated in Fig 2.5. These booms are composed of carbon fiber
reinforced polymer. In [23], the carbon fiber is Hexcel® IM7 and the polymer is
14
Case Out-of-Plane (Roll)
Torque (Nm)
In-Plane (Pitch/Yaw)
Torque (Nm)
Solar Cruiser
(1650 m2) 0° SIA
4.7×10-6 5.5×10-5
Solar Cruiser
(1650 m2) 35° SIA
4.9×10-5 2.4×10-3
Solar Polar Imager
(7000 m2) 0° SIA
7.7×10-5 4.6×10-3
Solar Polar Imager
(7000 m2) 35° SIA
6.4×10-4 4.7×10-2
Table 2.2: Alternative NASA requirements for attitude control of Solar Cruiser
and Solar Polar Imager [22].
Case Out-of-Plane (Roll)
Torque (Nm)
In-Plane (Pitch/Yaw)
Torque (Nm)
Solar Cruiser
(1650 m2) 0° SIA
4.7×10-6 5.5×10-4
Solar Cruiser
(1650 m2) 35° SIA
4.9×10-5 2.4×10-3
Solar Polar Imager
(7000 m2) 0° SIA
7.7×10-5 4.6×10-3
Solar Polar Imager
(7000 m2) 35° SIA
6.4×10-4 4.7×10-2
Table 2.3: Combined NASA requirements for attitude control of Solar Cruiser and
Solar Polar Imager.
15
977-2 resin. However, many different composite materials have been analyzed for
use as the material component for these three booms. The material properties of
this composite are listed in Table 2.4.
Figure 2.5: Cross-sectional models of three boom designs used in NASA solar sail
projects. Note that the dimensions can vary depending on the project [23].
IM7/977-2 Unitape Pre-preg
E11 (GPa) E22 (GPa) v12 v23 G12 (GPa)
173 9.17 0.34 0.37 5.65
Table 2.4: Material properties for Hexcel® IM7/977-2 [23].
The mass density of Hexcel® IM7/977-2 for a similar solar sail prototype
is recorded as 1780 kg/m3 [24]. It should be noted that these parameters are
16
Figure 2.6: Dimensions of the TRAC boom used in NASA solar sail projects [23].
significantly dependent on the composite material construction, and this is an
estimate using the parameters disclosed from several different sources and designs.
The TRAC boom design provided in Fig. 2.5 and Fig. 2.6 is modeled in Solid-
works. This is used to determine the centroid and the area moment of inertia in
order to approximate the axis of deformation and the magnitude of deformation
of the TRAC boom for simulation. The cross-section of this model is shown in
Fig. 2.7. The area moment of inertia values are given in Table. 2.5.
Moments of Inertia of the Area, at the Centroid (mm4)
Lxx = 717.57738 Lxy = 0.0 Lxz = 0.0
Lyx = 0.0 Lyy = 491.53865 Lyz = 0.0
Lzx = 0.0 Lzy = 0.0 Lzz = 1209.11602
Table 2.5: Area Moment of Inertia properties of the TRAC boom cross-section as
modeled in Solidworks [23].
The beam is modeled in simulation as a hollow disk with outer radius equal
to the distance from the centroid to the top of the web, and inner radius equal
to the distance from the centroid to the bottom of the web. This resolves to an
outer radius of 11.39486 mm and inner radius of 5.04486 mm. It is assumed that
17
Figure 2.7: Model of the TRAC boom cross-section in Solidworks [23].
the deformation occurs in the longitudinal direction, thus the elastic modulus of
E11, Poisson’s ratio of v12, and the shear modulus of G12 from Table 2.4 is used
to determine the magnitude of deformation.
It is noted that the dimensions given in Fig. 2.6 are over-constrained. Fur-
ther, different choices of minimum-constrained dimensions from Fig. 2.6 result in
slightly different cross-section models, which results in different area moments of
inertia. It is also noted that the area moment of inertia calculated in Table 2.5
using the dimensions given Fig. 2.6 are slightly different than the area moment
of inertia given in [23]. In fact, the dimensions chosen to constrain the model
in Fig. 2.7 are chosen such that the area moment of inertia was closest to the
value given in [23]. Based on these results, it appears that the dimensions given
in Fig. 2.6 have some error. However, since these values are only used to ap-
proximate the necessary tension forces to achieve a particular deformation, and
the error does not appear to be significant, these parameters given by [23] are
18
still used for the simulation. In particular, the moment of inertia and material
property values given by [23] are used in conjunction with the approximated outer
radius of 11.39486 mm and inner radius of 5.04486 mm.
The boom lengths are calculated using the mission criteria of 1650 m2 sail
area and 7000 m2 sail area for Solar Cruiser and Solar Polar Imager, respectively.
Assuming the sail is a perfect square, and thus each triangular component of the
sail between two booms is a 45-45-90 triangle, the boom lengths can be calculated
using the equation
L =
√
A√
2
, (2.3)
where L is the boom length and A is the sail area. This resolves to a boom length
of ∼28.72 m and ∼59.16 m for Solar Cruiser and Solar Polar Imager, respectively.
These are rounded to 29 m and 59 m for simplicity.
It is established that the SRP force acting on a sail at 1 astronomical unit
(AU) can be approximated using the expression
F ≈ ηPA cos2(α), (2.4)
where η ≈ 1.8 is the overall sail thrust coefficient, P = 4.563× 10−6 N/m2 is the
SRP constant at 1 AU, A is the solar sail area, and α is the SIA[25]. However, to
analyze the properties of a particular material, a more complete force model must
be defined. This model is derived from the assumption that photons can either
be absorbed, specularly reflected, or diffusely reflected, such that
1 = ρa + ρs + ρd, (2.5)
where ρa is the fraction of photons that are absorbed, ρs is the fraction of photons
that are specularly reflected, and ρd is the fraction of photons that are diffusely
reflected [25]. This is known to be a valid assumption for Aluminum front-coated
sails, where photon transmission drops to less than 1% for Aluminum coating of
19
40 nm [26]. Then the expression for the SPR force can be modeled as
F = Fnn+ Ftt, (2.6)
where Fn is the normal component of the SRP force, n is the unit vector normal
to the surface of the sail, Ft is the transverse component of the SRP force, and t
is a unit vector transverse to the surface of the sail [25]. The expression for the
normal force and the transverse force are given as
Fn = PA[(1 + ρs) cos
2(α) +
2
3
ρd cos(α)], (2.7)
Ft = PA(1− ρs) cos(α) sin(α). (2.8)
Different sail materials also have particular optical and thermal characteristics
that can affect the SRP force on the sail. Typical sails also use different materials
on the front and the back in order to optimize for reflectivity and emissivity,
respectively. Considering these factors, the expressions for the normal component
and the transverse component of the SRP force are
Fn = PA[(1+ rs) cos
2(α)+Bfr(1−s) cos(α)+ efBf − ebBb
ef + eb
(1− r) cos(α)], (2.9)
Ft = PA(1− rs) cos(α) sin(α), (2.10)
where r is the reflectivity of the front surface, s is the specular reflection coefficient,
Bf and Bb are the non-Lambertian coefficients for the front and back surfaces, and
ef and eb are the front and back surface emission coefficients [25]. The material
properties of the sail for NEA Scout are published and are used in the force
analysis for this thesis. The NEA Scout sail is made of 2.5 µm CP1 Polyimide
coated with 100 nm Al on the front and uncoated on the back [27]. The material
properties are given in Table 2.6.
Finally, the cable radius of the CDCM is selected to be 2.0 cm for both Solar
Cruiser and Solar Polar Imager. This is slightly less than double the radius of
20
Coefficient r s Bf Bb ef eb
Value 0.88 0.89 0.79 0.67 0.025 0.27
Table 2.6: Material properties for for the NEA Scout Sail [27].
the boom from the centroid to the tip of the boom web. Ideally, this cable ra-
dius should be as small as possible in order to integrate easily with the existing
NASA boom designs. However, this comes with a trade-off of larger cable tension
requirements, and thus larger power draw. To illustrate this, the lateral boom
deflection is plotted with respect to the applied cable tension for particular cable
radii and is shown in Fig. 2.8. From these plots, it is clear that a larger cable
radius results in a larger lateral boom deflection for a particular cable tension
setpoint.
(a) (b)
Figure 2.8: Lateral boom deflection vs. cable tension for the chosen disk radii of
2 cm, 4 cm, 6 cm, 8 cm, and 10 cm. The plots include the boom deflection for (a)
Solar Cruiser and (b) Solar Polar Imager.
21
2.5 Simulation Results
NASA requires a design margin such that the theoretical maximum attitude con-
trol torque is double the torque requirements outlined in Table 2.3, the designated
maximum operating point of the sail. This allows for more flexibility in the solar
sail operation and a larger margin for error. Reflecting these requirements, the
simulation is run to determine the minimum beam deflection to achieve double
the torque requirements.
2.5.1 Pitch/Yaw Torque Simulation Results
This simulation conducts a sweep across a minimum and a maximum cable tension
to determine the lateral beam deflection for each cable tension. This is done
by breaking the beam up into 580 finite elements that describe the position of
equally spaced points along the beam. For each simulated boom deflection, the
sail is broken into trapezoids, for which each trapezoid’s contribution to the overall
torque generation is calculated. The trapezoids are defined with height equal to
the difference between consecutive spaced points, and side lengths equal to the
sail length in the direction of the non-deflected boom, or normal to the deflected
boom, at the chosen spaced points. Similar to the work completed by the senior
design team, this analysis utilized the constant curvature subsegment (CCsub)
method for analyzing the deformation of the beam.
The clock angle is an important consideration when quantifying the magnitude
of attitude control that is feasible for the sail. In this thesis, clock angle is defined
as the rotation of the sun about the normal axis of the sail with respect to the
deflected beam. In particular, a clock angle of zero degrees indicates the sun is
directly aligned with the beam axis of deformation, while a clock angle of ninety
degrees indicates the sun is directly perpendicular to the beam axis of deforma-
tion. The analysis is conducted for clock angles of zero degrees and ninety degrees
since these clock angles indicate the orientations that provide the maximum and
minimum attitude control authority for an ideal sail. In this case, it is assumed
22
the sail has no effects of billowing or thermal expansion, and the sail deforma-
tion along the beam axis of deformation is equal to the magnitude of the beam
deformation. However, care should be taken when considering non-ideal sails, or
when operating in ways not outlined in this thesis (such as applying torsional and
lateral deformation simultaneously). In addition, further analysis is conducted for
stabilization control only, rather than full attitude control. For stabilization con-
trol, the design goal is to cancel any disturbance in the system and actuate the sail
orientation back to equilibrium. These disturbances could be from many sources,
but two of the most likely are non-idealities in the sail and unwanted bending or
flexing of the boom. Since the maximum theoretical actuation is achieved when
the sun is at a clock angle of zero degrees, it is assumed that the maximum dis-
turbance also occurs when the sun is aligned with this axis. Thus, for this case,
only a clock angle of zero degrees is analyzed.
Pitch/Yaw Simulation Results for Attitude Control
Utilizing these changes to the model, the torque generated with respect to cable
actuation for the TRAC boom is plotted in Figs. 2.9 and 2.10. These figures
illustrate the pitch/yaw torque generated with respect to cable actuation for Solar
Cruiser and Solar Polar Imager, respectively. Each of these plots assumes 20
distinct beam element disks for cable routing.
Notably, the torque generated in Figs. 2.9a, 2.9b, 2.10a, and 2.10b assume only
a single boom deformation at a time, whereas Figs. 2.9c and 2.10c assume both
booms along the same axis deform at the same angle. This is because, for 0° SIA
and for 35° SIA and 90° clock angle, torque generation for two booms of equal
deformation would cancel out. However, when the SIA is described along the axis
of deformation, i.e. for 0° clock angle, the torque generated can be increased by
deforming both booms along the same axis. In particular, the ideal orientation
of the two booms to achieve maximum torque generation would have one boom
oriented normal to the SRP, such that its effective SIA is reduced to zero, while
the other is deformed such that the boom is aligned with the SRP. For 0° SIA or
23
(a) (b)
(c)
Figure 2.9: Pitch/Yaw torque generated from lateral boom deformation for Solar
Cruiser with a 2.0 cm cable radius and 85 N of maximum tension. The plots
include the torque generated at (a) 0° SIA, (b) 35° SIA at 90° clock angle, and (c)
35° SIA at 0° clock angle.
24
(a) (b)
(c)
Figure 2.10: Pitch/Yaw torque generated from lateral boom deformation for Solar
Polar Imager with a 2.0 cm cable radius and 67 N of maximum tension. The plots
include the torque generated at (a) 0° SIA, (b) 35° SIA at 90° clock angle, and (c)
35° SIA at 0° clock angle.
25
when the SIA is described normal to the axis of deformation, both of the booms
are already normal to the SRP. Thus, only one boom should be deformed in order
to achieve maximum torque generation in this case.
The required lateral boom deformation to achieve double the torque require-
ments in Table 2.3 is also recorded. The simulation requires a cable tension
setpoint in order to generate the corresponding deformation, and this process is
not invertible. These results are instead recorded for a sweep of cable tension
setpoints, and the exact boom deformation required for the torque requirement
is obtained through linear interpolation of the results of the sweep. Thus, the re-
quired boom deformation to meet the torque requirements is close but not exact.
These results are recorded in Table 2.7.
Spacecraft Applied Cable
Tension (N)
Cable Length
Displacement
(m)
Boom
Displacement
(m)
Solar Cruiser 84.5198 0.0123 8.0087
Solar Polar
Imager
66.6000 0.0233 25.0591
Table 2.7: Cable tension, cable length displacement, and boom displacement re-
quired for attitude control to achieve double the NASA pitch/yaw torque require-
ments in Table 2.3.
The beam deformation required to achieve the maximum torques shown in
Figs. 2.9 and 2.10 is achieved by applying cable tensions of 85 N and 67 N,
respectively. The beam deformation for Solar Cruiser and Solar Polar Imager
that achieves these forces is illustrated in Fig. 2.11.
26
(a) (b)
Figure 2.11: Maximum beam deformation required for attitude control in Fig. 2.9
and Fig. 2.10. The plots include the deformation for (a) the Solar Cruiser boom
with 85 N of force and (b) the Solar Polar Imager boom with 67 N of force.
Pitch/Yaw Torque Simulation Results for Stabilization Control
Figs. 2.12 and 2.13 illustrate the beam deformation required if only stabilization of
the sail is desired. This assumes that the clock angle with the largest magnitude of
actuation is also the clock angle with the largest magnitude of disturbance forces.
Thus, only the zero clock angle is considered in this analysis. This simulation is
constructed using a TRAC boom cross-section with beam radius as in Fig. 2.7
and material properties and area moment of inertia given in [23]. Each of these
plots assumes 20 distinct beam element disks for cable routing.
The required lateral boom deformation to achieve double the torque require-
ments in Table 2.3 is also recorded for the stabilization control simulation. As
before, these results are recorded for a sweep of cable tension setpoints, and the
exact boom deformation required for the torque requirement is obtained through
linear interpolation of the results of the sweep. Thus, the required boom defor-
mation to meet the torque requirements is close but not exact. These results are
recorded in Table 2.8.
The beam deformation required to achieve the maximum torques shown in
Figs. 2.12 and 2.13 is achieved by applying cable tensions of 34 N and 17 N,
27
(a) (b)
Figure 2.12: Pitch/Yaw torque generated from lateral boom deformation for Solar
Cruiser with a 2.0 cm cable radius and 34 N of maximum tension. The plots
include the torque generated at (a) 0° SIA and (b) 35° SIA at 0° clock angle.
(a) (b)
Figure 2.13: Pitch/Yaw torque generated from lateral boom deformation for Solar
Polar Imager with a 2.0 cm cable radius and 17 N of maximum tension. The plots
include the torque generated at (a) 0° SIA and (b) 35° SIA at 0° clock angle.
28
Spacecraft Applied Cable
Tension (N)
Cable Length
Displacement
(m)
Boom
Displacement
(m)
Solar Cruiser 32.4776 0.0045 3.1327
Solar Polar
Imager
15.8028 0.0046 6.3643
Table 2.8: Cable tension, cable length displacement, and boom displacement re-
quired for stabilization control to achieve double the NASA pitch/yaw torque
requirements in Table 2.3.
respectively. The beam deformation for Solar Cruiser and Solar Polar Imager
that achieves these forces is illustrated in Fig. 2.14.
(a) (b)
Figure 2.14: Maximum beam deformation required for stabilization control in
Fig. 2.12 and Fig. 2.13. The plots include the deformation for (a) the Solar
Cruiser boom with 34 N of force and (b) the Solar Polar Imager boom with 17 N
of force.
2.5.2 Roll Torque Simulation Results
For cables that are threaded helically, an applied cable tension induces boom
twist rather than lateral boom deflection. If each of the four booms is twisted
in an equal manner, this induces a roll torque to the spacecraft. Although the
simulation derived from [20] is not designed for torsional boom deformation, this
29
can be modeled using certain assumptions. The code developed by the Senior
Design team at the University of Minnesota assumes that each boom is rotated
by a particular angle. This model further assumes the sail is only connected to
a spreader at the tips of the booms as well as at the center of the spacecraft.
Thus, when the booms are twisted, the sail forms four triangles rotated about the
satellite’s normal axis. This sail deformation is analogous to a pinwheel shape.
However, one limitation of this model is the assumption of only three connection
points for each triangular sail component. The work of the Senior Design team was
expanded in a similar manner to the lateral deformation analysis. Rather than
assuming the sail is only connected at three points, it is instead assumed that
the sail is connected at a constant cable radius along the entire beam. Instead
of assuming the beam is rotated by a fixed amount, the beam is analyzed using
a fixed number of finite elements. It is further assumed that the beam’s rotation
angle increases linearly, and the rotation does not change between each finite
element. Thus, the sail is broken up into planar trapezoids, each of which induces
a nominal torque to the spacecraft. The results of both of these two methods are
shown.
Simplified Roll Torque Simulation Results
The simplified roll torque generation model assumes a boom twist that is equal
across the entire boom. Further, the sail is assumed to only be attached at the
end-effector of each beam. It is also noted that this torque is achieved by twisting
all four booms equally and in the same direction. The results of this roll torque
model are shown in Figs. 2.15 and 2.16.
The torque generated in Figs. 2.15 and 2.16 assumes a spreader length of
15 cm for Solar Cruiser and 60 cm for Solar Polar Imager. In both cases, this
means the spreader would have to be much larger than the radius of the cables
threaded along the beam. However, the cable radius could instead be increased if
continuity is desired between the disk radius and the spreader length. This would
also increase the potential pitch/yaw torque that can be generated.
30
(a) (b)
Figure 2.15: Roll torque generated from boom twist for Solar Cruiser with a 15 cm
spreader length. The plots include the torque generated at (a) 0° SIA and (b) 35°
SIA.
(a) (b)
Figure 2.16: Roll torque generated from boom twist for Solar Polar Imager with
a 60 cm spreader length. The plots include the torque generated at (a) 0° SIA
and (b) 35° SIA.
31
In addition, the particular boom twist that achieves double the NASA re-
quirement for roll torque generation is given in Table 2.9. Note that this value is
subject to the choice of spreader length, and can be increased or decreased as a
design parameter choice.
Spacecraft Boom Twist Needed (degrees)
Solar Cruiser 5.9
Solar Polar Imager 4.6
Table 2.9: Boom twist required to achieve double the NASA roll torque require-
ments in Table 2.3.
Trapezoidal Roll Torque Simulation Results
While the previous method assumed the sail is only connected at the central
satellite hub and the two closest beams, a real solar sail is typically attached to
the beams of the spacecraft at equally spaced intervals of a fixed distance. When
the beams are composed of individual CDCM robots, it seems appropriate to
follow a similar approach and attach the sail along the disk elements. As a result,
it is important to consider the intermediate rotation of the beam.
For this analysis, it is assumed the beam rotation increases linearly. To ac-
complish this, the beam is broken into 580 finite elements that describe the
rotation of equally spaced points along the beam. These results are shown in
Figs. 2.17 and 2.18.
In addition, the particular boom twist that achieves double the NASA re-
quirement for roll torque generation is given in Table 2.10. Note that this value
is subject to the choice of spreader length, and can be increased or decreased as
a design parameter choice.
32
(a) (b)
Figure 2.17: Roll torque generated from boom twist for Solar Cruiser with a 15 cm
spreader length using the finite element trapezoid method. The plots include the
torque generated at (a) 0° SIA and (b) 35° SIA.
(a) (b)
Figure 2.18: Roll torque generated from boom twist for Solar Polar Imager with
a 60 cm spreader length using the finite element trapezoid method. The plots
include the torque generated at (a) 0° SIA and (b) 35° SIA.
33
Spacecraft Boom Twist Needed (degrees)
Solar Cruiser 5.8
Solar Polar Imager 4.5
Table 2.10: Boom twist required to achieve double the NASA roll torque require-
ments in Table 2.3 using the trapezoidal method.
2.5.3 Propulsion Force Losses Discussion and Simulation
Results
It is important to note that as the solar sail boom deformation increases, the
propulsive force decreases proportionally. This is because the component of the
SRP force normal to the sail decreases. This decrease in propulsive force is plotted
with respect to cable actuation length in Figs. 2.19 and 2.20 for lateral deforma-
tions to the beam, and with respect to boom twist in Figs. 2.21 and 2.22 for
torsional deformations to the beam. In all cases, the loss in propulsive force is
relatively small. Overall, this is advantageous to this attitude control design since
the propulsive force losses are small within the mission criteria for both Solar
Cruiser and Solar Polar Imager.
Additional figures are included in Appendix A. In particular, the roll, pitch,
and yaw torques are calculated for a simulated CTM beam whose cross-section is
given in Fig. 2.5.
34
(a) (b)
(c)
Figure 2.19: Propulsion (normal) force with respect to lateral cable actuation for
Solar Cruiser. The plots include the propulsive force on the sail at (a) 0° SIA, (b)
35° SIA at 90° clock angle, and (c) 35° SIA at 0° clock angle.
35
(a) (b)
(c)
Figure 2.20: Propulsion (normal) force with respect to lateral cable actuation for
Solar Polar Imager. The plots include the propulsive force on the sail at (a) 0°
SIA, (b) 35° SIA at 90° clock angle, and (c) 35° at 0° clock angle.
36
(a) (b)
Figure 2.21: Propulsion (normal) force with respect to torsional cable actuation
for Solar Cruiser using the trapezoid method. The plots include the propulsive
force on the sail at (a) 0° SIA and (b) 35° SIA.
(a) (b)
Figure 2.22: Propulsion (normal) force with respect to torsional cable actuation for
Solar Polar Imager using the trapezoid method. The plots include the propulsive
force on the sail at (a) 0° SIA and (b) 35° SIA.
37
2.6 Closing Remarks
The results of this chapter confirm the feasibility of the CABLESSail concept.
It was shown that the CABLESSail concept was capable of providing double the
NASA mission requirements for all cases. It also provides bounds for the design
parameters of the CDCM. In particular, it provides the cable tension requirements
of 0 N to 85 N for the TRAC boom design. Finally, the simulation developed in
this chapter will be utilized in the following chapters for deriving the forward
kinematics of the CDCM as well as for developing models of the CABLESSail
prototype.
The results of this section are also limited in several ways. The propensity
for buckling was not modeled in this simulation, so it was not clear whether the
TRAC boom could support the loads derived in this chapter. Further, several as-
sumptions were made about the TRAC boom model as well as the sail properties.
It is understood that there are many different variations of the TRAC boom con-
cept, which means the analysis in this chapter was limited to the chosen iteration.
However, this chapter was structured in such a way that, given the same material
properties of a different sail and boom, the same analysis could be conducted on
a different model in order to derive the same parameters. This will be utilized in
the following chapters to model the prototype CDCM.
Chapter 3
Deriving the Forward Kinematics
of the Cable-Driven Continuum
Manipulator
3.1 Introduction
Forward kinematics is defined as the process of obtaining the pose of an end-
effector given the joint angles of a robot. For the CDCM, this definition can
be extended to describe the pose achieved given particular motor torques and/or
cable lengths. Since the CDCM is a complex, nonlinear robot, deriving these
kinematics is particularly difficult. However, the forward kinematics of the CDCM
are a necessary component of the development of the CABLESSail prototype.
Several methods exist to compute the forward kinematics accurately, but these
methods are typically computationally expensive and are therefore not feasible
to run in real-time in a low resource environment such as a satellite. Therefore,
methods must be derived to approximate the forward kinematics accurately and
efficiently.
Three different approaches are implemented to derive this relationship for the
38
39
CDCM. The first approach is a simple polynomial fit function. The second ap-
proach is the equilibrium solution to a cantilever beam derived using the assumed
modes method. The final approach implements a neural network to learn this
relationship. In each of these three approaches, the beam is modeled after the
TRAC boom in [23].
3.2 Polynomial Fitting for the Forward
Kinematics of the CDCM
The simplest method for finding the end-effector pose of the CDCM is to solve the
end-effector solutions given a sequence of cable tension set points. Then, finding
the polynomial of best fit for this data results in a function that can be used to
easily predict the end-effector pose for any cable tension. To generate this data,
the code developed by [20] is again used with the beam properties of the TRAC
boom cross-section and zero gravitational acceleration. The beam of the CDCM is
modeled according to the Solar Cruiser design. Specifically, the beam is modeled
as a 29 m TRAC boom, the beam radius is modeled from Fig. 2.7 and the material
properties and area moment of inertia are given from [23]. The cable radius is
selected to be 2 cm. Each of these plots assumes 20 distinct beam element disks
for cable routing.
Unlike the simulations in Chapter 2, the polynomial fitting method in this
section utilizes a pseudo-rigid body method of analysis. This method is used
to more accurately model compliant/flexing bodies. This was also implemented
by [20] for CDCM deformation analysis, and their code is modified for the analysis
in this section. For the CDCM, the pseudo-rigid body analysis should return more
accurate results but generally takes longer to compute this result. This is ideal
for polynomial fitting, where the polynomial function should be as accurate as
possible. Further, the longer computation time is not a drawback as once the
polynomial function is fitted to the data, the model is no longer needed.
40
In order to find the optimal polynomial of best fit for the forward kinematics of
the CDCM, two design characteristics must be considered. The first is the order
of the polynomial. In general, a higher-order function better fits to existing data,
but generalizes worse to new data. This is the phenomenon known as over-fitting.
In order to avoid over-fitting, this analysis derives polynomials of orders 1 through
6, and compares the results to additional randomly generated data from the same
pseudo-rigid body model. The second is the method of generating data for curve
fitting. The most straightforward method is to simply generate uniformly spaced
data between the bounds of interest. However, an alternative method is found by
utilizing Chebyshev nodes, or the roots of the Chebyshev polynomials of the first
kind. Each of these results is applied to 500 randomly generated data points, and
the results of each of these methods are compared.
3.2.1 Polynomial Fitting for the Forward Kinematics of
the CDCM using Uniformly Spaced Data
As discussed, the most straightforward method for generating data for polynomial
fitting is to generate uniformly spaced data within the desired interval. As shown
in Section 2.5.1, the maximum cable tension needed is 85 N, so the polynomial
function is fitted to a single cable CDCM that can operate between 0 N and 85 N.
Each data point is spaced at 0.25 N intervals between 0 N and 85 N. The pose is
represented as a 7 index array, including the position and the quaternion of the
end-effector of the boom. The cable length is also recorded in each simulation
result. The change in cable length is thus calculated by subtracting the known
cable length at equilibrium from the cable length calculated in each simulation
iteration. Finally, the sign of the change in cable length is flipped to ensure a
positive change in cable length corresponds to a positive end-effector displacement.
The independent variables can be either the cable tension or the change in cable
length. This results in a total of 14 different polynomial functions. However, since
the boom is only deflected along a single axis, one of the position axes and two of
41
the quaternion axes remain zero for any boom deflection and cable tension. This
reduces the total number of polynomials to 8 for each polynomial order. The R2
values of each of these functions, as well as the root mean square error (RMSE) of
the function applied to random data, are shown in Tables 3.1 and 3.2, respectively.
Polynomial Order
Function Order 1 Order 2 Order 3 Order 4 Order 5 Order 6
x1 (m) vs. cable
tension (N)
0.9985 1.0000 1.0000 1.0000 1.0000 1.0000
x1 (m) vs.
∆ cable
length (m)
0.9981 1.0000 1.0000 1.0000 1.0000 1.0000
x3 (m) vs. cable
tension (N)
0.9073 0.9993 1.0000 1.0000 1.0000 1.0000
x3 (m) vs.
∆ cable
length (m)
0.9494 0.9998 1.0000 1.0000 1.0000 1.0000
qw vs. cable
tension (N)
0.9055 0.9992 1.0000 1.0000 1.0000 1.0000
qw vs. ∆ cable
length (m)
0.9480 0.9998 1.0000 1.0000 1.0000 1.0000
qy vs. cable
tension (N)
0.9978 1.0000 1.0000 1.0000 1.0000 1.0000
qy vs. ∆ cable
length (m)
0.9987 1.0000 1.0000 1.0000 1.0000 1.0000
Table 3.1: R2 values of the polynomial fit function applied to uniformly spaced
data.
The results in Tables 3.1 and 3.2 show that the polynomial fit functions gen-
erated from uniformly spaced data are effective at approximating the equilibrium
position and rotation of the boom. As the R2 value of a fit function is a representa-
tion of its goodness-of-fit, the values of near 1 for all trials indicate a near-perfect
goodness-of-fit. It is also clear that nearly all trials showed better results by se-
lecting the change in cable length as the independent variable rather than cable
42
Polynomial Order
Function Order 1 Order 2 Order 3 Order 4 Order 5 Order 6
x1 (m) vs. cable
tension (N)
0.1046 0.0029 0.0009 0.0001 0.0000 0.0000
x1 (m) vs.
∆ cable
length (m)
0.1197 0.0015 0.0004 0.0000 0.0000 0.0000
x3 (m) vs. cable
tension (N)
0.1962 0.0175 0.0011 0.0000 0.0000 0.0000
x3 (m) vs.
∆ cable
length (m)
0.1463 0.0103 0.0004 0.0000 0.0000 0.0000
qw vs. cable
tension (N)
0.0052 0.0005 0.0000 0.0000 0.0000 0.0000
qw vs. ∆ cable
length (m)
0.0039 0.0002 0.0000 0.0000 0.0000 0.0000
qy vs. cable
tension (N)
0.0044 0.0001 0.0000 0.0000 0.0000 0.0000
qy vs. ∆ cable
length (m)
0.0034 0.0000 0.0000 0.0000 0.0000 0.0000
Table 3.2: RMSE values of the polynomial fit function generated from uniformly
spaced data applied to randomly generated data.
43
tension. This is advantageous because the cable length is easily measured by an
encoder attached to a motor that actuates on the cables. The polynomial of or-
der 3 is selected as it is the lowest order polynomial with an R2 value of 1.0000.
This keeps the function relatively simple while still retaining a high degree of
accuracy. The exact polynomial functions are given in Table 3.3.
Function Equation
x1 (m) vs. ∆ cable length (m) x1 = −2.0611 × 104l3 − 6.3121 × 103l2 +
726.6426l − 0.0011
x3 (m) vs. ∆ cable length (m) x3 = 1.4596 × 105l3 − 1.1632 × 104l2 −
1.4949l + 29.0011
qw vs. ∆ cable length (m) qw = 3.4242×105l3−298.2675l2−0.0457l+
1.0000
qy vs. ∆ cable length (m) qy = 427.1131l
3 − 204.8282l2 + 25.0148l −
1.0116× 10−5
Table 3.3: Functions representing the equilibrium solution for the CDCM pose vs.
cable length.
Although the polynomial function is designed to have either the change in cable
lengths or the cable tensions as the independent variable, the PRB model only
accepts cable tensions as an input. Thus, the cable lengths are generated from
the output of this model, and may not follow a uniform distribution. Despite this,
the cable lengths polynomials performed better on the randomly generated data
points. Thus, the validity of the cable lengths polynomial function is confirmed.
3.2.2 Polynomial Fitting for the Forward Kinematics of
the CDCM using Chebyshev Nodes
The Chebyshev nodes, also known as the roots of the Chebyshev polynomials
of the first kind, are a sequence of data that minimizes Runge’s Phenomenon.
Runge’s Phenomenon is the problem of poor polynomial fitting, often in the form
of oscillations and often near the extrema of the data. This is illustrated in
Fig. 3.1.
44
Figure 3.1: Illustration of Runge’s Phenomenon, and the advantage of Chebyshev
interpolation (CIP) over other interpolation methods. In this case, Lagrange
interpolation (LIP) [28].
It is proven that for n data points in the range (−1, 1), the Chebyshev nodes
produce the unique polynomial of order n that minimizes the error within these
bounds. Given the interval (−1, 1), and a number of nodes n, the Chebyshev
nodes are defined as
xk = cos (
2k − 1
2n
π), k = 1, 2, ..., n. (3.1)
This can be extended to any interval using an affine transformation. This results
in a modified definition for the Chebyshev nodes. For the data in the interval
(a, b), the modified Chebyshev nodes equation is given by
xk =
1
2
(a+ b) +
1
2
(b− a) cos (2k − 1
2n
π), k = 1, 2, ..., n. (3.2)
For this analysis, the same PRB model is used as in the previous section.
This means the beam is modeled according to the TRAC boom and is operated
between cable tensions of 0 N and 85 N. The same number of nodes are used
in this analysis as in the uniformly spaced nodes analysis in order to maintain
continuity. In total, this means there are 341 nodes. The Chebyshev nodes are
45
generated according to Eq. (3.2), where the bounds are the cable tension limits
of 0 N and 85 N. As before, the change in cable length is calculated for each
simulation. Both cable tensions and change in cable length are analyzed using
a polynomial fit. The R2 values of each of these functions, as well as the root
mean square error (RMSE) of the function applied to random data, are shown in
Tables 3.4 and 3.5, respectively.
Polynomial Order
Function Order 1 Order 2 Order 3 Order 4 Order 5 Order 6
x1 (m) vs. cable
tension (N)
0.9986 1.0000 1.0000 1.0000 1.0000 1.0000
x1 (m) vs.
∆ cable
length (m)
0.9982 1.0000 1.0000 1.0000 1.0000 1.0000
x3 (m) vs. cable
tension (N)
0.9151 0.9994 1.0000 1.0000 1.0000 1.0000
x3 (m) vs.
∆ cable
length (m)
0.9540 0.9998 1.0000 1.0000 1.0000 1.0000
qw vs. cable
tension (N)
0.9135 0.9993 1.0000 1.0000 1.0000 1.0000
qw vs. ∆ cable
length (m)
0.9528 0.9998 1.0000 1.0000 1.0000 1.0000
qy vs. cable
tension (N)
0.9980 1.0000 1.0000 1.0000 1.0000 1.0000
qy vs. ∆ cable
length (m)
0.9988 1.0000 1.0000 1.0000 1.0000 1.0000
Table 3.4: R2 values of the polynomial fit function applied to data generated from
the Chebyshev node distribution.
The results in Tables 3.4 and 3.5 show that the polynomial fit functions gener-
ated from the Chebyshev node distribution are also effective at approximating the
equilibrium position and rotation of the boom. As the R2 value of a fit function
is a representation of its goodness-of-fit, the values of near 1 for all trials indicate
a near-perfect goodness-of-fit. As with the uniformly distributed data simulation,
46
Polynomial Order
Function Order 1 Order 2 Order 3 Order 4 Order 5 Order 6
x1 (m) vs. cable
tension (N)
0.1222 0.0035 0.0010 0.0001 0.0000 0.0000
x1 (m) vs.
∆ cable
length (m)
0.1408 0.0018 0.0005 0.0001 0.0000 0.0000
x3 (m) vs. cable
tension (N)
0.2310 0.0211 0.0013 0.0001 0.0000 0.0000
x3 (m) vs.
∆ cable
length (m)
0.1712 0.0122 0.0005 0.0000 0.0000 0.0000
qw vs. cable
tension (N)
0.0061 0.0006 0.0000 0.0000 0.0000 0.0000
qw vs. ∆ cable
length (m)
0.0045 0.0003 0.0000 0.0000 0.0000 0.0000
qy vs. cable
tension (N)
0.0052 0.0001 0.0000 0.0000 0.0000 0.0000
qy vs. ∆ cable
length (m)
0.0040 0.0000 0.0000 0.0000 0.0000 0.0000
Table 3.5: RMSE values of the polynomial fit function generated from the Cheby-
shev node distribution applied to randomly generated data.
47
nearly all trials showed better results by selecting change in cable length as the in-
dependent variable rather than cable tension. As before, the polynomial of order 3
is selected as it is the lowest order polynomial with an R2 value of 1.0000. This
keeps the function relatively simple while still retaining a high degree of accuracy.
The exact polynomial functions are given in Table 3.6.
Function Equation
x1 (m) vs. ∆ cable length (m) x1 = −2.0420 × 104l3 − 6.3048 × 103l2 +
726.4877l − 6.0815× 10−4
x3 (m) vs. ∆ cable length (m) x3 = 1.4560 × 105l3 − 1.1635 × 104l2 −
1.3612l + 29.0006
qw vs. ∆ cable length (m) qw = 3.4157×103l3−298.4230l2−0.0415l+
1.0000
qy vs. ∆ cable length (m) qy = 427.7693l
3 − 204.7419l2 + 25.0132l −
4.3566× 10−6
Table 3.6: Functions representing the equilibrium solution for the CDCM pose vs.
cable length using the Chebyshev node distribution.
Although the polynomial function is designed to have either the change in cable
lengths or the cable tensions as the independent variable, the PRB model only
accepts cable tensions as an input. Thus, the cable lengths are generated from the
output of this model, and may not follow a Chebyshev node distribution. Both
choices of independent variables are tested using 500 randomly generated data
points. As the polynomial functions using the change in cable lengths had both
higher R2 and lower RMSE values, the accuracy of the cable length polynomial
function is confirmed.
3.2.3 Polynomial Fitting Discussion
Based on the results of the polynomial fitting for both the uniformly distributed
data and the Chebyshev nodes, one might conclude that the uniformly distributed
interpolation polynomials perform better than the Chebyshev interpolation poly-
nomials. Although the Chebyshev interpolation polynomials in general had better
48
R2 values, the RMSE values of the uniformly distributed interpolation polynomials
are smaller. This would seem to indicate that the uniformly distributed interpola-
tion polynomials would perform better on unseen data. However, it is shown that
this result is somewhat misleading. An extension of the Chebyshev node func-
tion is proposed, which is used to generate random data according to the same
distribution. This is shown in Eq. (3.3).
xk =
1
2
(a+ b) +
1
2
(b− a) cos (u(x)
2
π), (3.3)
where u(x) is the continuous uniform random distribution between (0,1), with
probability density function given by
u(x) =
1 for x ∈ (0, 1)0 otherwise . (3.4)
Sampling from this continuous uniform distribution yields data that is distributed
between (a,b) according to the Chebyshev distribution extended to continuous
time i.e. with infinite nodes. This data is then used in the PRB model to generate
equilibrium pose solutions. This is useful as an indicator of the accuracy of the
polynomial fit functions with preference towards the extrema.
Generating 500 random data using the sampling method in Eq. (3.3), the
RMSE values of the uniformly distributed and Chebyshev interpolation polyno-
mials are compared for the polynomials of orders 1 to 3. These results are shown
in Table 3.7.
The RMSE values calculated from the Chebyshev random distribution show
better performance for the Chebyshev interpolation polynomials compared to the
uniform random distribution. Furthermore, the RMSE values for the Chebyshev
interpolation in Tables 3.5 and 3.7 are approximately the same, whereas the RMSE
values for the uniformly distributed interpolation in Table 3.7 are larger than any
other RMSE data. This indicates that the uniformly distributed interpolation
49
Uniform Data Chebyshev Data
Function Order 1 Order 2 Order 3 Order 1 Order 2 Order 3
x1 (m) vs. cable
tension (N)
0.1418 0.0042 0.0013 0.1290 0.0036 0.0011
x1 (m) vs.
∆ cable
length (m)
0.1626 0.0021 0.0006 0.1488 0.0018 0.0005
x3 (m) vs. cable
tension (N)
0.2683 0.0249 0.0016 0.2450 0.0217 0.0013
x3 (m) vs.
∆ cable
length (m)
0.1977 0.0146 0.0006 0.1807 0.0126 0.0005
qw vs. cable
tension (N)
0.0071 0.0007 0.0001 0.0065 0.0006 0.0000
qw vs. ∆ cable
length (m)
0.0052 0.0003 0.0000 0.0048 0.0003 0.0000
qy vs. cable
tension (N)
0.0060 0.0001 0.0000 0.0055 0.0001 0.0000
qy vs. ∆ cable
length (m)
0.0047 0.0000 0.0000 0.0043 0.0000 0.0000
Table 3.7: RMSE values of the polynomial fit function generated from the Cheby-
shev random distribution.
polynomials are better at predicting data near the center of the distribution but
worse at predicting data near the extrema. In contrast, the similar RMSE values
for the Chebyshev interpolation generated from uniform random data and from
the Chebyshev random distribution indicates that the Chebyshev interpolation
provides the same accuracy across the entire data range - in this case, between
0 N and 85 N of cable tension. For these reasons, the Chebyshev interpolation is
concluded to be the more accurate polynomial fit.
Although the polynomial function is shown to have very accurate results when
compared to the equilibrium pose solutions, this method does have some draw-
backs. First, the implementation only considers a single cable, whereas a real
CDCM would have multiple cables actuating simultaneously. Further work would
50
be needed to determine if the polynomial interpolation is still valid for a multi-
variable input. Second, the polynomial only finds the equilibrium solution to the
pose. This neglects the dynamics of the CDCM. As a result, using this solution
in a control system may impact performance or stability.
3.3 Assumed Modes Method for the Forward
Kinematics of the CDCM
The assumed modes method is a numerical analysis method used to approximate
the solution to a forced vibration problem using a series of basis functions [29].
These functions multiplied by the time-dependent solution to the assumed modes
model provide an approximate solution to the continuous system. Explicitly, the
approximate solution to the one-dimensional continuous system is given by
u(x, t) =
n∑
i=1
Ψi(x)qi(t) = Ψ(x)q(t). (3.5)
The equations of motion for a cantilever beam can be derived using the expression
given in Eq. (3.5) [30]. This derivation solves for the potential and kinetic energy
of the system, then uses the Euler-Lagrange equation to describe the force applied
to the beam. The Euler-Lagrange equation is defined as
fB =
d
dt
(
∂LB
∂q˙
)⊤ − (∂LB
∂q
)⊤, (3.6)
where
LB = TB − VB. (3.7)
Here, Eq. (3.7) is the Lagrangian with kinetic energy TB and potential energy
VB. The fB term is computed as the reaction force on the beam due to the cable
51
tension. Using Eq. (3.6), the equations of motion for the boom are derived as
MBq¨+KBq = fB. (3.8)
In Eq. (3.8), MB is the mass matrix of the system, and KB is the stiffness matrix.
The analysis in [30] also includes a damping coefficient to ensure the beam does
not sustain oscillations and reaches equilibrium. This is added to the equations
of motion, resulting in the full expression as
MBq¨+ DBq˙+KBq = fB, (3.9)
where DB is the damping matrix containing the damping coefficients of the system.
The equilibrium solution of the equations of motion can be easily computed
by finding the value of q for q¨ = q˙ = 0. This solution can be converted to the
beam deflection by plugging into Eq. (3.5). The equilibrium solution to the end-
effector of the beam is defined as ueq. Although the problem is still nonlinear, this
simplifies the problem significantly, albeit at the loss of the transient response.
The beam and material properties are derived from the Solar Cruiser model
and [23], as in Chapter 2. This provides the boom length as well as the components
of the mass matrix and the stiffness matrix given in 3.9. The damping matrix is
selected to be the diagonal matrix of 7×103. This is chosen so the pole frequency
remains approximately the same, but the solution eventually reaches equilibrium.
The results of the assumed modes plot for the end-effector of the beam, along with
the equilibrium solution, are shown in Fig. 3.2. This result assumes an applied
cable tension of 2 N. For these chosen parameters, the equilibrium solution is
found to be 1.8450 m of deflection.
While this results in an equilibrium solution to the system, this result can be
further improved by finding the envelope of the oscillation decay. This can be
solved by finding a function of the form
y(t) = ueq + be
−ct. (3.10)
52
Figure 3.2: Cantilever beam simulation plotted over time along with its equilib-
rium position.
The coefficients b, c can be estimated by a logarithm transformation. This trans-
formation is given as
yL(t) = log(y(t)− ueq) = log(be−ct) = bL − ct. (3.11)
It is important to note that only values greater than zero are valid in the logarithm
function. Thus, only values for which y(t) > ueq are considered. Finally, the
problem can be solved using linear least squares as
min
bL,c
1
2
∥bL − ct− yL(t)∥22 such that bL − ct ≥ yL(t). (3.12)
Intuitively, this least squares function finds the coefficients for the linear function
that is closest to the data that is also greater than the data for all time t.
Using the data generated in Fig. 3.2, the coefficients bL and c are found to be
53
0.6159 and 0.0051, respectively. After transforming bL back to b, this results in
the envelope function given by
y(t) = 1.8450 + 1.8514e−0.0051t. (3.13)
This function only plots the positive envelope of the function. For completeness,
the negative envelope is given by
y(t) = ueq − be−ct. (3.14)
This function is solved using the same steps as the positive envelope function. For
the parameters given in Fig. 3.2, this results in the equation
y(t) = 1.8450 + 1.8514e−0.0050t. (3.15)
Because the oscillations may not reach the same peak value for both the negative
and the positive data, the more correct envelope is given by the envelope with
the largest bounds. In this case, since the negative envelope decays slower, it has
larger bounds. Thus, for the parameters given in Fig. 3.2, the correct envelope
function is
y(t) = 1.8450± 1.8514e−0.0050t. (3.16)
This function is plotted over the transient response data as shown in Fig. 3.3.
This result is advantageous because it provides bounds for the decay of the
end-effector position. This means that, for a particular cable tension, the error
bounds of the position can be given in addition to the equilibrium position. This
is extremely important for a number of algorithms, such as Kalman filtering, to
provide a filtered position approximation. However, it is more computationally
expensive than the assumed modes algorithm, as it requires a solution using the
assumed modes method in addition to solving the envelope function. As an ex-
tension of this work, it may be possible to approximate the coefficients bL and
c as a function of cable lengths or tensions. This would improve on the result
54
Figure 3.3: Cantilever beam simulation plotted over time along with its bounding
envelope.
from Section 3.2 as it would provide error bounds in addition to the equilibrium
solution.
It is noted that the code developed for use in the assumed modes method is
a work in progress for the CABLESSail project, and thus has several limitations.
The first and most significant is that the results do not match the results given
in [20]. Though both make use of the design and material properties of the beam,
the resulting equilibrium solutions do not match. This discrepancy would have
to be solved in order to move forward with either method. Second, the damping
matrix was selected arbitrarily rather than based on any data. This too would
need to be modeled in order to move forward with the assumed modes method for
a CABLESSail prototype. Despite these shortcomings, the results are extremely
promising as a computationally efficient method of deriving the transient response
of the boom.
55
3.4 Neural Network Method for the Forward
Kinematics of the CDCM
The neural network is the cornerstone of machine learning. It is constructed
from individual “neurons” which, based on their connection, attempt to produce
a desired result [31]. The most significant advantage of the neural network is
its flexibility, as the neural network can be tasked to learn nearly any problem.
However, neural networks may fail to achieve their desired behavior for a variety
of reasons. The challenge of neural network design, therefore, is constructing the
algorithm in such a way that the network accurately models the desired behavior.
The neural network is constructed from layers of perceptrons, each with a
particular activation function. Traditionally, a single perceptron is fed input data
from the previous layer. Each of these inputs is applied a weight coefficient, and
the sum of this data is computed. Finally, a non-linear activation function is
applied to this sum. An illustration of the perceptron is shown in Fig. 3.4. The
neural network is constructed from these individual perceptrons, where the output
of one perceptron serves as the input to the next layer. The general construction of
a neural network is illustrated in Fig. 3.5. This is also known as a single “neuron”
in a neural network.
Applying the neural network to the task of solving the forward kinematics of
the CDCM is analogous to fitting a non-linear function to the CDCM pose data
using regression. In the case of the neural network regression problem, the output
layer does not have an activation function; rather, it simply outputs a number
that represents the output of the learned function.
Training a neural network involves evaluating the network according to a loss
function. For a regression problem, the standard loss function is the half-mean-
squared-error loss function. This is given as
J(w|r, x) = 1
2
∥r− y∥22. (3.17)
56
Figure 3.4: Illustration of a standard perceptron with an arbitrary non-linear
activation function [32].
Figure 3.5: Illustration of a standard neural network with a single output [33].
57
In Eq. (3.17), J(w|r, x) represents the error (or loss) associated with the particular
weight coefficients, w, r represents the ground truth data, and y represents the
prediction output by the neural network.
The most standard form of neural network is the fully connected neural net-
work. In this construction, every neuron in a particular layer is connected to every
neuron in the following layer. In effect, every output of each neuron in one layer
is an input to every neuron in the following layer. For most machine learning
problems, this is the first attempted method used to learn a problem. For regres-
sion problems in particular, the fully connected neural network is sufficient for
accurately predicting the output of a problem. Thus, the fully connected neural
network is used as the model for learning the forward kinematics of the CDCM.
In general, the best activation function for a regression problem is the rectified
Linear Unit (ReLU) activation function [34]. This function is defined as
f(z) = max (0, z). (3.18)
In recent years, the ReLU activation function has become the most common choice
for activation function in standard machine learning problems [35]. The main
advantage of the ReLU activation function is it avoids the vanishing gradient
problem. The vanishing gradient problem is the phenomenon where, during neu-
ral network training, the gradient can become very small, leading to ineffective
learning in earlier layers of the network. Since the backpropagation step of train-
ing relies on the gradient of the loss function to improve the neural network, a
very small gradient causes the improvement to be ineffective. For this reason, the
ReLU activation function is used for each of the layers for learning the forward
kinematics of the CDCM.
In order to compute the training update functions, the gradients associated
with each weight must first be derived. These functions are given as
58
∂J(w|rtj, xt)
∂ytj
= −(rtj − ytj), (3.19)
∂ytj
∂wij,k
= xtik , (3.20)
∂ytj
∂xtik
= wij,k , (3.21)
∂xtij
∂wi−1j,k
=
xti−1k for x > 00 otherwise , (3.22)
∂xtij
∂xti−1k
=
wi−1j,k for x > 00 otherwise . (3.23)
Definitions for the parameters in Eqs. (3.19)-(3.23) are given in Table 3.8.
Symbol Description
y Output predicted by the neural network
r Ground-truth data
x Hidden or input node, connects to other
hidden nodes or the output
t Index of the data
i Layer of the neural network
j Index of the output node
k Index of the input node
Table 3.8: Parameters of the neural network gradient equations given in
Eqs. (3.19)-(3.23).
Using the equations derived in Eqs. (3.19)-(3.23), the weight update functions
can thus be expressed. For a neural network of arbitrary depth n, using the ReLU
activation function and the half-mean-squared-error loss function, the gradient
of a particular weight can be derived as
∂J(w|rtj ,xt)
∂wnl,m
using the chain rule. Finally,
the weights are updated during training using standard gradient descent, with
59
equation given by
wnl,m = wnl,m − η
∂J(w|rtj, xt)
∂wnl,m
. (3.24)
In Eq. (3.24), η is a tunable learning rate that determines how much the weights
change each iteration. The learning rate is set to 0.0001 from experimental obser-
vation.
While the overall structure of the neural network has been established, the
exact parameters to optimize the network for the task of predicting the forward
kinematics still need to be determined. For this analysis, neural networks with
depths of 1-6 are trained to determine the optimal layer depth. Further, each
neural network is tested with 10 and 100 nodes per layer. In addition, each of the
neural networks is trained on the same data multiple times. This is known as an
epoch. For each neural network, epochs of 1, 51, 101, 251, and 501 are tested.
Data was generated using the PRB function developed by [20] with parameters
selected according to the Solar Cruiser model and with material properties of the
TRAC boom as given by [23]. In this construction, the CDCM had two direct
threaded cables in order to provide lateral deformation in one axis. The inputs to
this system are the combined vector of cable lengths and cable tensions. As with
the model in Section 3.2, the outputs are the position and quaternion of the pose.
Two different methods are used to generate the four cable tensions. The first
method used a uniform random distribution on the interval (0 N, 85 N), which
was established as the necessary bounds of operation for the TRAC boom. The
second method used the Chebyhsev random distribution established by Eq.(3.3).
One million sets of four cable tensions are generated according to each of these
random distributions. Nine hundred thousand of these data points are used to
train the neural network, while one hundred thousand data points are used for
validation. These results are compared for each of the parameters.
Across all of these configurations, there are a total of 100 different neural
network constructions. The average RMSE value for each of these parameters is
provided. First, the average RMSE for each variation of epoch is recorded. This
60
is shown in Table 3.9. Next, the average RMSE for each variation of layers is
recorded. This is shown in Table 3.10. Following this, the average RMSE for each
variation of nodes per layer is recorded. This is shown in Table 3.11. Finally,
the neural networks generated according to the uniform random distribution and
the Chebyhsev distribution are compared. As with the results in Section 3.2,
each of the sets of neural networks can be compared to ground-truth data that
is generated according to the uniform random distribution and the Chebyshev
random distribution. In total, this provides four different average RMSE values.
These are given in Table 3.12.
Number of Epochs Average Pose RMSE
1 6.5219
51 1.0729
101 0.8350
251 0.6655
501 0.5205
Table 3.9: Average RMSE of the pose with respect to the number of epochs.
Number of Layers Average Pose RMSE
1 2.1337
2 3.1392
3 1.4182
4 1.4146
5 1.5102
Table 3.10: Average RMSE of the pose with respect to the number of layers.
Number of Nodes per Layer Average Pose RMSE
10 2.5421
100 1.3042
Table 3.11: Average RMSE of the pose with respect to the number of layers.
61
Validation Data Distribution
Training Data
Distribution
Uniform Random
Validation Data
Chebyshev Random
Validation Data
Uniform Random
Training Data
2.0507 2.1697
Chebyshev Random
Training Data
1.6880 1.7844
Table 3.12: Average RMSE of the pose with respect to the method of data gen-
eration.
It is observed that the neural networks that were trained with fewer epochs
had bad overall performance. Thus, these results are affected by the random ini-
tialization of the network more than neural networks that were trained with more
epochs. To get a more accurate comparison of the performance of the Chebyshev
random and the uniform random neural networks, the average RMSE is recorded
for the number of layers, number of nodes per layer, and method of data sampling,
but only considering neural networks with training epochs of 101, 251, and 501.
These results are shown in Tables 3.13, 3.14, and 3.15.
Number of Layers Average Pose RMSE
1 0.4661
2 2.0593
3 0.2152
4 0.2498
5 0.3779
Table 3.13: Average RMSE of the pose with respect to the number of layers,
considering only epochs 101, 251, and 501.
Number of Nodes per Layer Average Pose RMSE
10 1.2315
100 0.1158
Table 3.14: Average RMSE of the pose with respect to the number of layers,
considering only epochs 101, 251, and 501.
62
Validation Data Distribution
Training Data
Distribution
Uniform Random
Validation Data
Chebyshev Random
Validation Data
Uniform Random
Training Data
0.6667 0.7490
Chebyshev Random
Training Data
0.6166 0.6623
Table 3.15: Average RMSE of the pose with respect to the method of data gen-
eration, considering only epochs 101, 251, and 501.
From the average RMSE data using epochs of 101, 251, and 501, it is clear that
certain parameters have a positive effect on the accuracy of the neural network.
It is clear that increasing the number of epochs of training, increasing the number
of nodes per layer, and using the Chebyshev random training data have a positive
effect on the accuracy of the neural network. It is also clear that increasing
the number of layers also increases the accuracy of the neural networks up to 3
layers, but has the opposite effect for layers 4 and 5. This is likely due to over-
fitting, where the neural network has fitted itself to the training data and has lost
generalizability. Each of the 100 combinations of parameters is also compared
to determine the overall neural network construction with optimal performance.
Interestingly, the network with the lowest average RMSE utilized 5 layers, 100
nodes per layer, 501 epochs, and the Chebyshev random distribution for generating
data. This network construction had an average RMSE value of 0.0428 across
both the Chebyshev and uniform random validation data. This disagrees with
the results of Table 3.13, which identified 3 layers as the optimal construction.
This could be due to numerous factors, but is most likely caused by the network
getting stuck in local minima. Overall, however, it appears the optimal network
construction utilizes 3 layers, 100 nodes per layer, 501 epochs, and the Chebyshev
random distribution for generating data.
One of the major advantages of the neural network is its overall flexibility. As
a consequence, the neural network could be modified to be much more general,
63
such as to predict the forward kinematics of any flexible beam rather than just
the TRAC boom. This would require the parameters of the particular boom to be
added as inputs to the neural network and significant data on the ground-truth
data for this beam. Furthermore, the neural network is able to consider as many
or as few inputs as desired. In this construction, the neural network considers both
the cable tensions and cable lengths, providing more information on the current
state of the CDCM. As an extension of this work, the neural network could be
modified to consider the material properties in order to predict the deformation
of other beam models.
3.5 Closing Remarks
Three different methods were derived to approximate the forward kinematics of the
CDCM. Each of these three methods has particular advantages and disadvantages.
The polynomial fitting method yielded surprisingly accurate results despite its
relatively simple construction. However, in its current construction, it could only
predict the CDCM beam deformation due to the actuation of a single cable.
Compared to this, the assumed modes method could also predict the transient
response of the beam. However, it was more computationally expensive than
the polynomial method and could also only predict the deformation due to the
actuation of a single cable. Finally, the neural network method could predict
the equilibrium position of the beam deformation due to the actuation of two
parallel cables. This could be further generalized for any number of additional
cables or other parameters entirely, including for the material properties of the
beam. However, it has lower accuracy than the polynomial fitting method despite
requiring more data, and has no information on the transient response.
Each of these three methods has areas that could be improved, which could al-
leviate some of these drawbacks. For example, both the polynomial fitting method
and the assumed modes method could be expanded for use with multiple cables.
64
Additionally, the envelope of the assumed modes method could be predicted us-
ing either the polynomial fitting method or the neural network method to reduce
computation time. Alternatively, the neural network method could be trained
further to reduce the RMSE error. Overall, it is clear that the forward kinematics
can be approximated using one of several techniques. However, the best approach
will depend on the particular limitations of a design.
Chapter 4
Hardware Design of the
Cable-Actuated Bio-inspired
Lightweight Elastic Solar Sail
4.1 Introduction
In order to develop the full-scale CABLESSail prototype, a prototype CDCM is
designed for future integration as the flexible beam elements of the CABLESSail
concept. Therefore, the concept of the CDCM is implemented in hardware to
provide a proof of concept and validation of the models developed in Chapter 2.
In this section, three iterations of the CDCM design are discussed. The particular
hardware selected for use in these designs is also motivated. Finally, the experi-
mental results are demonstrated along with a comparison to the results predicted
by the model in Chapter 3. These include the change in motor rotation, the in-
duced load torque from the deflected beam, and the end-effector displacement of
the flexible beam. This provides evidence of the feasibility of the CABLESSail
concept along with a qualitative analysis of the actuator requirements for the
CDCM.
65
66
4.2 Design
The design of the CDCM consists of three iterations. The first and second iter-
ations of the CDCM model are constructed with support from Michael Dallalah
and Michael States, two undergraduate Aerospace Engineering and Mechanics
students at the University of Minnesota. The third iteration of the CDCM model
is constructed with support from Michael States. The first iteration is a general
proof of concept with an oversized base and oversized disk beam elements to en-
sure a successful result. This iteration only tested cable tensions that resulted in
large lateral deformation. The second iteration focused on decreasing the cable
radius to more accurately model the ratio of the beam length to the cable radius
that would be required on a real solar sail. It also tested torsional deformation
to determine feasibility with respect to the NASA mission criteria. The final it-
eration focused on improvements to the structural design, with a minor revision
to the beam elements. Each iteration of the CDCM structure is designed and
modeled in Solidworks. These are then exported to STL Files and 3-D printed us-
ing a Prusa i3 MK3S+. This method allowed for flexibility in the design process,
including ease of part replacement and allowing the use of unique or non-standard
part structures.
4.2.1 Iteration I
The CDCM base is constructed from four modular pieces, with an additional
insert for the base of the boom. Each of these four modular components attaches
to a motor enclosure and enclosure plate. The four modular pieces are connected
with a bolt. In addition, the base insert is bolted to the four modular pieces.
This iteration uses a 3/8 in diameter polycarbonate beam. The isometric model
is shown in Fig. 4.1.
The first iteration of the CDCM utilizes beam elements with 2 in cable radius
with respect to the central beam. This design utilized four beam elements includ-
ing the end cap. The isometric model of the beam elements and the end cap is
67
Figure 4.1: Isometric model of the first iteration of the CDCM base.
shown in Fig. 4.2.
(a) (b)
Figure 4.2: Isometric model of the first iteration of the CDCM beam elements.
This includes (a) the standard beam elements and (b) the beam end cap.
One of the main drawbacks of this design is that the beam is mounted below
the motors. This means the motors are not actuating on the entire beam, which
reduces their control authority. In addition, since the beam insert is below the
motors, the first beam element moves relative to the base frame as the motors
actuate on the cables. This is undesirable as it increases friction and is inaccurate
68
with respect to standard CDCM models, including the one used in Section 2.4.
Additional figures are included in Appendix B.1. In particular, an engineering
diagram of the base and an additional isometric model is included for further
design clarity.
4.2.2 Iteration II
The second iteration modified the beam elements to have a 0.5 in cable radius.
This cable radius is chosen to demonstrate the capability of deformation for a
CDCM with a much tighter cable radius, which more strongly resembles the di-
mensions of a real solar sail boom. This design again used four beam elements
including the end cap. The isometric model of the beam elements and the end
cap is shown in Fig. 4.3. The second iteration of the CDCM used the same base
design and polycarbonate beam as the first iteration.
(a) (b)
Figure 4.3: Isometric model of the second iteration of the CDCM beam elements.
This includes (a) the standard beam elements and (b) the beam end cap.
4.2.3 Iteration III
The third iteration of the CDCM included a redesign of the base and added
support for several variations of beams. This design is constructed using a combi-
nation of 3-D printed components and other construction materials. The frame is
69
constructed from four 9 in t-slotted rails. These are connected by two 3-D printed
bases. The enclosure plate and the spool supports are mounted to the frame us-
ing t-slot fasteners. The motors, enclosures, and spools are vertically staggered to
eliminate the additional friction caused by the cables bending outward from the
last beam element to the spool. Each of the spools is supported by ball bearings.
The isometric model is shown in Fig. 4.4.
Figure 4.4: Isometric model of the third iteration of the CDCM base.
This design supports modular swapping of the beam insert. This means that
the same base frame can be used for any beam, as the insert can be unbolted and
swapped with a different insert that supports a different beam. In addition, this
ensures the cables form a normal vector with respect to the base plate. This is im-
portant as it ensures the cable position coincident with the frame does not change
as the motor actuates on the cable and the cable length increases or decreases.
Two different inserts are designed for this iteration of the CDCM. The first is
compatible with a 3/8 in polycarbonate beam as is used in every iteration of the
design. The second insert supports a 1 in tape measure TRAC boom. These are
illustrated in Fig. 4.5 and 4.6.
The tape measure beam insert is constructed in two parts. The tape measure
70
Figure 4.5: Isometric model of the polycarbonate beam insert for the third itera-
tion of the CDCM.
(a) (b)
Figure 4.6: Isometric model of the tape measure beam insert for the third iteration
of the CDCM. This includes (a) the insert for the 1 in tape measure TRAC boom
and (b) the wedge for the 1 in tape measure TRAC boom.
71
TRAC boom is first inserted into the base plate, shown in Fig. 4.6a. This plate
is designed to fit the flange of the boom. Each of two wedges, shown in Fig. 4.6b,
is then inserted to hold the tape measure TRAC boom in place.
The beam elements are also modeled in Solidworks and 3-D printed to fit either
the tape measure TRAC boom or the polycarbonate beam. The beam elements for
the polycarbonate beam are slightly modified from the second iteration. Note that
the end cap is unchanged from the second iteration of the CDCM. The isometric
models for these beam elements are shown in Fig. 4.7.
(a) (b)
Figure 4.7: Isometric model of the third iteration of the CDCM beam elements.
This includes (a) the standard beam elements and (b) the beam end cap.
Whereas the beam elements for the polycarbonate beam closely resemble the
disks of a conventional CDCM, the beam elements of the tape measure TRAC
boom are small cable routings that are mounted along each axis of the boom. The
isometric model of these beam elements is given in Fig. 4.8. These TRAC boom
elements are designed to allow the beam elements to be mounted to the TRAC
boom without impeding its stowed configuration. This design allows cables to be
routed directly along the axis of the flange for lateral deformation of the boom,
and helically through all three beam elements for torsional deformation. However,
this design does not allow reverse lateral deformation, unlike the polycarbonate
beam.
The dimensions of the tape measure TRAC boom are modeled in Solidworks
72
Figure 4.8: Isometric model of the beam elements used for the tape measure
TRAC boom.
using the measured dimensions of the assembled beam. The cross-sectional model
is shown in Fig. 4.9. Notably, the path length is 25.40 mm (1.0 in), and the
thickness is 0.127 mm (0.005 in). These dimensions are used to find the area
moment of inertia in Solidworks. The area moment of inertia values are given in
Fig. 4.1.
Moments of Inertia of the Area, at the Centroid (mm4)
Lxx = 242.91932 Lxy = 0.0 Lxz = 0.0
Lyx = 0.0 Lyy = 157.15591 Lyz = 0.0
Lzx = 0.0 Lzy = 0.0 Lzz = 400.07523
Table 4.1: Area moment of inertia properties of the tape measure TRAC boom
cross-section.
In simulation, the beam is modeled as a hollow disk with outer radius equal
to the distance from the centroid to the top of the web, and inner radius equal to
the distance from the centroid to the bottom of the web. This resolves to an outer
radius of 11.63393 mm and inner radius of 6.81559 mm. Furthermore, the cable
radius is determined as the distance from the centroid to the outer web plus the
radius of the cable routing hole given in Fig. 4.8. The radius of the cable routing
hole is 0.125 in, or 3.175 mm, which yields a total cable radius of 14.80893 mm
(0.5830287 in). As discussed in Section 4.2.2, the disk beam elements for the
polycarbonate beam have a 0.5 in cable radius.
73
Figure 4.9: Model of the tape measure TRAC boom cross-section.
Additional figures are included in Appendix B.2. In particular, an engineering
diagram of the base and additional isometric models of the base and the TRAC
boom are included for further design clarity.
4.3 Hardware Selection
Hardware is selected based on flexibility and ease of integration with these designs.
As the ARDC lab at the University of Minnesota has significant experience with
using ODrive Robotics hardware, this is selected as the basis for hardware design.
In particular, the ODrive Pro is selected for use as the motor controller for each of
the four motors on the CDCM. The ODrive Pro is a high performance hobby motor
controller that has significant customizability. It allows for position control, torque
control, and velocity control, which makes it flexible for all possible use cases in
this preliminary CDCM design. The ODrive motor controller software provides a
74
feedback loop with the motors and encoder to match the desired position, torque,
or velocity with the measured value. The ODrive D6374 brushless DC motor is
selected since it provides simple integration with the ODrive Pro motor controller.
These motors can also provide up to 3.86 Nm of torque [36]. Noting that the spools
have a 0.25 in radius, this means these motors can provide∼600 N of cable tension.
The AMT212B-V-OD absolute encoder is selected to provide encoder feedback to
the ODrive Pro. This is an AMT212B-V absolute encoder with custom ODrive
firmware to enable higher speeds and less jitter. The ODrive Pro, D6374 motor,
and AMT212B-V-OD encoder are illustrated in Fig. 4.10.
The encoder feedback is used as the measurement for an outer feedback con-
troller that supplies inputs to all four motor controllers. The Arduino Giga R1 is
selected as the central processor for the encoder data and outer loop controller.
The four ODrive Pros are connected via TTL UART to the four Serial ports on
the Arduino. The ODrive Arduino library, provided by ODrive [37], is modified
to read the encoder data and provide the desired output. The Arduino Giga R1
is shown in Fig. 4.11.
Since the ODrive motor with a 0.25 in radius winch is capable of providing up
to 600 N of cable tension, the cables are selected to be 250 Lb rated Kevlar line.
This is equivalent to ∼1112 N of tension, approximately double the maximum
tension that the motor and winch combination can produce. These cables also
have a high strength to mass ratio and significantly less stretch compared to nylon
wire.
As discussed in Section 4.2, the two beams used in prototyping are a poly-
carbonate beam and a tape measure TRAC boom. The polycarbonate beam is
a Grainger 3 ft by 3/8 in general purpose polycarbonate rod. The tape measure
TRAC boom is constructed from a Stanley 25 ft by 1 in steel tape measure. This
tape measure is made from a C1095 steel alloy and hardened to R15N 83-86. Two
different sizes of TRAC boom are chosen - one 1 m boom and the other 2.5 m
boom. To construct the tape measure TRAC boom, two pieces of the desired boom
length are cut from the tape measure. These pieces are taped together to replicate
75
(a) (b)
(c)
Figure 4.10: Hardware selection for the CDCM. This includes (a) the ODrive
Pro motor controller, (b) the ODrive D6374 brushless DC motor, and (c) the
AMT212B-V-OD absolute encoder.
76
Figure 4.11: Arduino Giga R1 used for CDCM control
the flange of the TRAC boom. The beam elements also provide additional sup-
port by holding the two tape measure components together. The polycarbonate
beam and the tape measure TRAC boom are shown in Figs. 4.12 and 4.13. In
addition, the fully assembled third iteration of the CDCM with the polycarbonate
beam is shown, along with the CDCM with a nominal deflection, in Fig. 4.14.
(a) (b)
Figure 4.12: Polycarbonate beam for the CDCM.
To provide power to the ODrives, two 24 V, 600 W Mean Well power supplies
are selected for AC/DC power conversion and power supply. Each of these power
supplies provides power to two ODrives, which in turn provides power to a single
77
(a) (b)
Figure 4.13: Tape measure TRAC boom for the CDCM. Shown is the 1 m tape
measure TRAC boom.
(a) (b)
Figure 4.14: Fully assembled third iteration of the CDCM. Shown are (a) the
CDCMwith no applied cable tension and (b) the CDCMwith a nominal deflection.
78
motor. The line between the power supply and the ODrive is connected by a
SureStep regen clamp, which is rated for 7 A and 50 W continuous, 800 W peak
power. The power supply and regeneration clamp are shown in Fig. 4.15.
(a) (b)
Figure 4.15: (a) The SureStep regeneration clamp and (b) the Mean Well LRS-
600-24 power supply.
Noting that the torque constant of the ODrive D6374 motor is ∼0.053 Nm/A,
the maximum safe torque that can be provided by the motors is 0.371 Nm. Re-
calling that the spool radius is 0.25 in, this means the maximum safe cable tension
that can be supplied by the motors is 58.425 N. However, it is important to note
that the motor is capable of exceeding the safe bounds of the regeneration clamp.
In future iterations, the regeneration clamp may need to be changed in order to
provide wider safety margins.
The entire parts list is included in Appendix C for reference.
4.4 Results
Testing is conducted on the CDCM with the assistance of Michael States and Niko
Sexton, two undergraduate Aerospace Engineering and Mechanics students at the
University of Minnesota. The CDCM is configured for use with the polycarbonate
79
beam for testing in this section. The ODrive is first set to a zero position where
the cables are taut but not pulling. The ODrive is then set to filtered position
control and set to positions between 0.1 rotations and 0.8 rotations in 0.1 rotation
increments. The results are recorded in radians, which correspond to a minimum
set point of 0.6283 radians, a maximum of 5.0265 radians, and 0.6283 radian
increments. Three pairs of Vicon markers are placed along the CDCM beam, in
addition to four markers placed in the beam insert, in order to track the position
of the CDCM. The first pair of markers are placed about 6 inches from the base,
the second near the center of the beam, and the third about 4 inches from the tip
of the beam. These markers are recorded using Vicon MX cameras and Tracker
3.9 software [38]. The motor rotation, angular velocity, and torque applied are
also recorded in Matlab, using the UART TTL connection from the Arduino Giga
to the ODrive Pros and the USB Serial connection from the computer to the
Arduino Giga. The polycarbonate beam is also modeled in simulation, and the
experimental and simulation results for the end-effector displacement and motor
torque are compared. While the tape measure TRAC booms are not tested in
hardware, the 2.5 m tape measure TRAC boom is modeled in simulation.
The simulations developed in Chapter 2 are modified to reflect the material
properties of the polycarbonate beam and the tape measure TRAC boom. The
material properties for each of these materials are found online [39–41]. The ma-
terial properties of polycarbonate are estimated as the parameters in Table 4.2.
The material properties of C1095 steel alloy are estimated as the parameters in
Table 4.3.
Polycarbonate Material Properties
E (GPa) v (Poisson’s Ratio) G (GPa) Mass Density
(kg/m3)
2.378 0.36 0.786 1.2×103
Table 4.2: Material properties for polycarbonate [39, 40].
80
C1095 Steel Alloy Material Properties
E (GPa) v (Poisson’s Ratio) G (GPa) Mass Density
(kg/m3)
205 0.29 80.0 7.85×103
Table 4.3: Material properties for a C1095 steel alloy [41].
The simulation model of the boom used for these results uses the PRB model.
This is chosen so the results of the simulation are as accurate as possible to
compare with the experimental results. However, it is important to note that,
while the gravitational acceleration in the PRB model is changed to reflect Earth’s
gravity for this analysis, the resulting beam deformation predicted by simulation
is found to be the same for any gravitational acceleration value. It appears that
there is a bug in the code for the PRB model developed by [20], where this model
does not consider gravity in the beam deformation model.
The polynomial interpolation technique is modified from Chapter 3 for use
in this analysis. Two equations are derived using the polynomial interpolation
method: the first to describe the CDCM end-effector displacement with respect
to the load torque, and the second to describe the load torque with respect to
motor rotations. Noting the winch radius of 0.25 in, the motor rotations can be
calculated using the cable length displacement predicted by the simulation. In
a similar way, the load torque can be calculated using the winch radius as well
as the cable tension predicted by the simulation The data is generated using the
Chebyshev nodes given by Eq. (3.2) with 801 data nodes.
For the polycarbonate beam simulation, the sample tension forces are selected
to be in the range of (0 N, 200 N). For the tape measure TRAC boom simulation,
the sample tension forces are selected to be in the range of (0 N, 2500 N). In
each case, these ranges are found to actuate the cables of the simulated CDCM
up to 0.8 rotations. In this simulation, the cable actuation and the displacement
of the end-effector are significantly larger than in Section 3.2. From running the
simulation, it is observed that the third order polynomial is no longer sufficient
81
to accurately fit the data. Thus, a sixth order polynomial is selected to fit the
data from this simulation. This sixth order polynomial is generated using the
Chebyshev nodes and has an R2 value of 1.0000 for both polynomial functions.
The equations of best fit are given in Tables. 4.4 and 4.5.
Function Equation
x1 (m) vs. load torque (Nm) x1 = 0.0779T
6
l − 0.1305T 5l − 0.0550T 4l −
0.2694T 3l +0.1710T
2
l +0.8688Tl+6.1969×
10−6
load torque (Nm) vs.
∆ motor position (rads)
Tl = −1.7857×10−7p6+6.5142×10−6p5−
1.1776 × 10−4p4 + 0.0016p3 − 0.0195p2 +
0.2630p+ 1.6679× 10−6
Table 4.4: Functions representing the equilibrium solution for the polycarbonate
beam displacement and motor torque vs. rotations.
Function Equation
x1 (m) vs. load torque (Nm) x1 = −3.0023×10−9T 6l +1.1430×10−6T 5l −
1.9810 × 10−5T 4l − 5.5827 × 10−4T 3l +
0.0014T 2l + 0.2261Tl − 2.3580× 10−6
load torque (Nm) vs.
∆ motor position (rads)
Tl = −1.5305×10−7p6+5.6619×10−6p5−
8.6102 × 10−5p4 + 0.0016p3 − 0.0568p2 +
2.3723p− 5.6853× 10−7
Table 4.5: Functions representing the equilibrium solution for the tape measure
TRAC boom displacement and motor torque vs. rotations.
The predicted cable tension and end-effector position for both the polycarbon-
ate beam and tape measure TRAC boom simulations and each of the set points
are provided in Tables 4.6 and 4.7. The results of the rotation of the motor, the
applied motor torque, and the end-effector deflection are plotted in Fig. 4.16.
There are several limitations of the data generation from the Vicon cameras
that should be noted. First, the data generated from the Vicon cameras had to be
synchronized manually. To do this, the time coinciding with the final position at
the set point is offset until they match with the time of the trial for the 0.8 rotation
82
Motor
Actuation (rads)
Tip
Deflection (mm)
Torque
Required (Nm)
0.6283 140.3526 0.1579
1.2566 270.2868 0.3025
1.8850 385.1375 0.4356
2.5133 481.6619 0.5586
3.1416 557.9021 0.6729
3.7699 613.0756 0.7793
4.3982 647.4245 0.8788
5.0265 662.0315 0.9722
Table 4.6: Simulation results for the 3 ft polycarbonate beam using the PRB
model.
Motor
Actuation (rads)
Tip
Deflection (mm)
Torque
Required (Nm)
0.6283 333.1305 1.4685
1.2566 651.2663 2.8943
1.8850 944.0947 4.2793
2.5133 1202.7734 5.6252
3.1416 1420.3351 6.9336
3.7699 1591.8906 8.2060
4.3982 1714.6700 9.4437
5.0265 1787.9477 10.6480
Table 4.7: Simulation results for the 2.5 m tape measure TRAC boom using the
PRB model.
83
(a) (b)
(c)
Figure 4.16: Experimental results for the CDCM with the polycarbonate beam.
This includes (a) the rotation of the motor, (b) the applied motor torque, and (c)
the displacement of the end-effector.
84
(5.0265 rads) set point. Second, the zero point for the Vicon data is set based
on the average position at equilibrium. This nonzero equilibrium is calculated
to be 30.34 mm, and indicates that the beam may not be perfectly straight.
Qualitatively, the beam does appear to have some natural bend. Finally, the end-
effector position is calculated using a straight-line approximation between the final
two pairs of Vicon markers. These markers are located near the center of the beam
and about 4 in from the tip of the beam. Thus, there is some error between the
straight-line approximation and the true displacement of the tip of the beam.
For the motor rotation data, the rotation also had to be offset by a nonzero
initial position. This is because the motor calculates the relative position based
on the startup position. This initial position is determined to be 0.075 rotations;
however, this initial position offset would likely change if these experiments are
repeated, and simply represents the absolute position calculated at startup. When
moving to the full-scale CABLESSail prototype, it may be useful to define an
absolute position to ensure the positioning of the motor is consistent.
The equilibrium end-effector position, motor position, and applied torque are
calculated for each of the set points as well. This is found by taking the average
value between the time interval 12 s and 14 s, which is observed to be well after the
transience has decayed. In particular, the time constant of the filtered position
function in ODrive is known to be 0.5 s, so this coincides with 24 to 28 time
constants beyond the initial time. At this point, the output can be considered to
be at equilibrium. These equilibrium positions are recorded in Table 4.8.
The results in Tables 4.6 and 4.8 indicate the model compares very poorly
to the experimental results. Although the motor actuation matches closely be-
tween simulation and hardware, the tip deflection and the torque requirements
are significantly lower in hardware compared to simulation. This is an interesting
result because it indicates the model incorrectly assumes the relationship between
motor rotation and end-effector position. The polynomial fit between the cable
torque and tip deflection vs. motor rotations assumed there is no cable stretch-
ing; however, a nominal cable stretch could result in the results seen in Table 4.8.
85
Motor
Actuation (rads)
Tip
Deflection (mm)
Torque
Required (Nm)
0.6282 15.20 0.0235
1.2564 45.17 0.0294
1.8847 95.65 0.1793
2.5130 162.17 0.2142
3.1410 240.23 0.2871
3.7694 323.97 0.3665
4.3977 401.09 0.4298
5.0260 478.89 0.6430
Table 4.8: Experimental equilibrium results for the 3 ft polycarbonate beam.
Thus, rather than attempting to match the motor actuation, the simulation is run
using the cable tensions corresponding to the applied motor torques observed in
experimentation. These results are recorded in Table 4.9.
Motor
Actuation (rads)
Tip
Deflection (mm)
Torque
Required (Nm)
0.0899 20.5162 0.0235
0.1127 25.6934 0.0294
0.7179 159.6427 0.1793
0.8666 191.1275 0.2142
1.1871 256.5697 0.2871
1.5524 326.4640 0.3665
1.8565 380.3217 0.4298
2.9729 539.4855 0.6430
Table 4.9: Simulation results for the 3 ft polycarbonate beam using the PRB
model and the experimental torques for each set point.
The results in Table 4.9 are much closer to the experimental results for the
tip deflection given in Table 4.8. While the tip deflection matches closer with
the experimental results, the motor rotation has decreased significantly compared
to the experimental results. This provides further evidence that the cables are
stretching. While these results still have some error, it is noted that these mate-
rial properties are found online for general purpose polycarbonate. It is further
86
noted that polycarbonate, and plastics in general, have a high variance in material
properties. As an extension, it would be useful to determine the properties of this
polycarbonate beam. Alternatively, testing the tape measure may provide better
results since the hardened steel should have more accurate material properties. It
may also be possible to reduce the cable stretching by selecting a different cable
material. Any of these would be useful to improve the accuracy of the model and
determine the accuracy of the PRB model as a whole.
4.5 Closing Remarks
The concept of the CDCM was implemented in hardware using a polycarbonate
beam and a tape measure TRAC boom to represent the flexible boom of the CA-
BLESSail design. Over three iterations, the design was improved to reduce the
size of the CDCM while also implementing the design constraints of the CAB-
LESSail prototype. The prototype was also tested at a sequence of set points.
These results were compared to results predicted by the simulations developed
in the previous chapters. Initially, the simulation results compared poorly to the
experimental results. However, it was shown that this was likely due to cable
stretching.
Because of the disparity between the predicted and experimental results, sig-
nificant work will be needed to improve the simulation model. As discussed, some
of the most straightforward methods to improve this model would be to replace
the cables with less flexible cables, determine the material properties of the poly-
carbonate beam experimentally, or replace the beam with a material with more
consistent material properties. However, these were not the only disadvantages
of the model. This model could only predict the equilibrium position of the end-
effector position. For the full-scale CABLESSail prototype, it will be important
to determine the damping of the beam in order to determine how long it takes to
reach equilibrium.
87
Despite these limitations, the experimental results show promise for the feasi-
bility of the CABLESSail concept. The CDCM was able to effectively deflect the
beam in excess of the requirements needed for the full-scale Solar Cruiser design.
In future work, the most important aspect to improve in the design would be the
forward kinematics of the CDCM. This will become extremely important when
estimating the position of the end-effector and controlling the deformation to a
particular pose.
Chapter 5
Modeling the Dynamics of the
Cable-Driven Continuum
Manipulator Hardware
5.1 Introduction
Thus far, the derived models for the CABLESSail design have focused on the
system in static equilibrium. However, it is important to model the dynamics of
the system to ensure stability and quantify the bounds of operation, including the
oscillations, actuation bounds, and power requirements. Therefore, the CDCM
hardware is modeled using a high-fidelity PMSM model. The controller for this
motor is developed from the field-oriented control technique. This model is simu-
lated using the Solar Cruiser TRAC boom model, the tape measure TRAC boom
model, and the prototype polycarbonate beam. The results of the polycarbonate
beam simulation are also compared to experimental results.
88
89
5.2 Permanent Magnet Synchronous Motor
Concepts
A permanent magnet synchronous motor (PMSM) is an alternating current (AC)
synchronous motor that uses a wire-wound stator to create a magnetic field and
rotate the permanent magnet rotor. The PMSM shares many properties with the
brushless direct current (BLDC) motor. In particular, they both have a wire-
wound stator, permanent magnet rotor, and are electronically commutated mo-
tors. However, the main difference is the sinusoidal counter-electromotive force
(back-EMF) produced by the windings in a PMSM, as opposed to the trapezoidal
back-EMF of the BLDC motor. This back-EMF waveform is a property of the
geometry of the motor, and cannot be controlled or modified by the controller.
The most fundamental parameters of any motor are the phase resistance, phase
inductance, speed constant, and torque constant. These parameters define the
overall performance of the motor. The phase resistance and phase inductance are
simply the equivalent resistance and inductance that exist in each phase line of
the motor. However, sometimes these parameters are given phase-phase and not
phase-neutral, so care must be applied when interpreting these parameters. The
speed constant defines the ratio between the mechanical velocity of the motor
and the voltage applied. Typically, this ratio is given in rpm
VLL,pk
. In order to
translate this line-line voltage to the line-neutral voltage, the voltage is divided
by
√
3. This assumes the voltages are sinusoidal and are 120° out of phase, as is
typical of most PMSMs. Finally, the torque constant defines the ratio between
the electromechanical torque output and the current. This ratio is typically given
in Nm
Apk
. However, it can also be derived from the speed constant. This relationship
is given as
KT =
3
2
1√
3
60√
2π
1
KV
. (5.1)
In this expression, each of the coefficients has a particular interpretation. The first
90
coefficient, 60√
2π
, represents a change of units from rpm to rad
s
. The second coeffi-
cient, 1√
3
, converts the voltage from line-line to line-neutral. The final coefficient,
3
2
, represents the total torque generated from the contribution of all three currents.
This torque constant is derived assuming a sinusoidal current commutation, which
is valid for most PMSMs.
To optimize the performance of the motor, the applied current should match
the waveform of the back-EMF. This is evident by the equation for the motor
torque, which is given by
Te =
eaia + ebib + ecic
ωm
, (5.2)
where ej is the back-EMF of winding j, and ωm is the motor velocity in
rad
s
[42].
By convention, the motor velocity is defined in the counter-clockwise direction.
Since the back-EMF in a PMSM is sinusoidal, the per-phase current should also be
sinusoidal in phase with the back-EMF. The generalized ideal per-phase current
waveform is given by
Ia = −Ipk sin(ωet), (5.3)
Ib = −Ipk sin(ωet− 2π
3
), (5.4)
Ic = −Ipk sin(ωet− 4π
3
). (5.5)
Similarly, the per-phase back-EMF waveform is given by
ea = − ωm√
3KV,rad
sin(ωet), (5.6)
eb = − ωm√
3KV,rad
sin(ωet− 2π
3
), (5.7)
ec = − ωm√
3KV,rad
sin(ωet− 4π
3
). (5.8)
Here, ωm is the rotor mechanical velocity and ωe is the electrical velocity, both
in rad
s
. The electrical velocity is given as the frequency of the stator voltages,
91
which depends on the number of magnet pole pairs. The
√
3 is included since
the back-EMF is defined for the line-neutral voltage, but KV is defined for the
line-line voltage.
Using Eq. (5.2), the torque can be calculated as
Te =
ωmIpk(sin
2(ωet) + sin
2(ωet− 2π3 ) + sin2(ωet− 4π3 )
ωm
√
3KV,rad
= Ipk
3
2
1√
3
1
KV,rad
= IpkKT .
(5.9)
This shows that when the current and back-EMF have the same waveform, the
torque output by the motor is constant.
From this simple torque calculation, it is clear that the motor dynamics are
difficult to calculate due to the three-phase sinusoidal waveforms. Rather than
calculating the dynamics in the static frame of reference, the signals can be trans-
formed into the rotor’s rotating frame of reference. In this frame of reference, and
assuming a balanced system, the three-phase AC system can be modeled as if it is
a two-parameter DC system. This is known as field-oriented control (FOC). This
method has several other advantages as well, including reduced torque ripple and
higher motor efficiency, particularly at high speeds [43, 44]. This method relies
on two transformations: the Clarke transformation and the Park transformation.
This two-step process is illustrated in Fig 5.1.
In a three-phase motor, the three current signals are 120° out of phase. This
is shown in Eqs. (5.3)-(5.5). Assuming a balanced system, the Clarke transfor-
mation transforms these three phases into a static, orthogonal, two-axis reference
frame [46]. The Clarke transformation is defined as
Iα(t)
Iβ(t)
Iγ(t)
 = 23

1 −1
2
−1
2
0
√
3
2
−
√
3
2
1
2
1
2
1
2


Ia(t)
Ib(t)
Ic(t)
 . (5.10)
92
Figure 5.1: Illustration of the Clarke and Park transformation from a three-phase
static reference frame to a two-parameter rotating reference frame [45].
In a balanced three-phase system, Iα(t) = I
s
d(t), Iβ(t) = I
s
q (t), and Iγ(t) = 0, as
shown in Fig. 5.1.
After translating the three-phase system to a two-phase orthogonal system,
the Park transform applies a rotation matrix between the rotor and stator. This
translates the signals from the static reference frame of the stator to the rotating
reference frame of the rotor. The Park transform is given as
Id
Iq
Iz
 =

cos θ sin θ 0
− sin θ cos θ 0
0 0 1


Iα(t)
Iβ(t)
Iγ(t)
 . (5.11)
Combined, these two transformations apply a direct-quadrature-zero (DQZ) trans-
formation [47]. Intuitively, the DQZ transform is a technique to model a balanced
three-phase AC system as a two-phase DC system. In the two-phase DC system,
the two phases are given by the q-axis and d-axis current. The equivalent circuit
for the q-axis current is shown in Fig. 5.2, assuming the d-axis current is zero.
The DQZ transformation is applied to the ideal current equations given in
Eqs. (5.3)-(5.5). After applying the Clarke transformation, the resulting currents
are given as 
Iα(t)
Iβ(t)
Iγ(t)
 =

−Ipk sin(ωet)
Ipk cos(ωet)
0
 . (5.12)
93
e
q
L
s
R
s
V
q
I
q
Figure 5.2: Equivalent q-axis circuit diagram of a three-phase PMSM after apply-
ing the DQZ transformation, assuming zero d-axis current.
Using the currents derived from the Clarke transform, the DQZ currents are found
by applying the Park transform. These currents are found to be
Id
Iq
Iz
 =

0
Ipk
0
 . (5.13)
Thus, it is shown that the Clarke and Park transforms change the time-varying
sinusoidal current under ideal conditions to a single, constant parameter. In gen-
eral, the ideal operation of a PMSM motor controller is to drive Id to zero and Iq
to the desired line current.
The PMSM model is defined in the frame of reference of the rotor [48]. The
equations that model the dynamics of the PMSM are
94
Vd = IdRs +
dλd
dt
− ωeLqIq, (5.14)
Vq = IqRs +
dλq
dt
+ ωeLdId + ωeλpm, (5.15)
eq = ωeλpm, (5.16)
λd = LdId + λpm, (5.17)
λq = LqIq, (5.18)
Te =
3
2
p(λpmIq + (Ld − Lq)IdIq), (5.19)
Te − Tl = J dωm
dt
+Bf , (5.20)
Bf =

Bvωm +Bcsign(ωm) for ωm ̸= 0
Bs,maxsign(Te − Tl) for ωm = 0, |Te − Tl| ≥ Bs,max
Te − Tl for ωm = 0, |Te − Tl| < Bs,max
, (5.21)
ωm =
ωe
p
. (5.22)
The parameters of Eqs. (5.14)-(5.22) are defined in Table 5.1.
95
Symbol Description
Vd d-axis voltage (V)
Vq q-axis voltage (V)
eq q-axis back-EMF (V)
Id d-axis current (A)
Iq q-axis current (A)
Rs Stator phase resistance (Ω)
λpm Permanent magnet flux linkage (Weber)
λd d-axis flux linkage (Weber)
λq q-axis flux linkage (Weber)
ωe Electrical velocity i.e. frequency of
stator voltages ( rad
s
)
ωm Rotor mechanical velocity (
rad
s
)
Ld d-axis inductance (Henry)
Lq q-axis inductance (Henry)
Te Electromechanical torque produced by
the PMSM (Nm)
Tl Load torque (Nm)
p Pole pairs
J Motor moment of inertia (kg-m2)
Bf Motor friction
Bv Viscous friction coefficient (Nm-s)
Bc Coulomb friction coefficient (Nm)
Bs,max Static friction coefficient (Nm)
Table 5.1: Parameters of the PMSM equations given in Eqs. (5.14)-(5.22) [48].
96
5.3 Modeling the ODrive D6374
The prototype CDCM developed in Chapter 4 utilized ODrive hardware for the
motor and motor controller. The particular motor selected for this prototype was
the ODrive D6374 motor. This is a three-phase, high torque, outrunner hobby
motor. An example of an outrunner hobby motor is shown in Fig. 5.3. This figure
Figure 5.3: Interior of the KDE Direct KDE2315XF-885, a hobby-grade outrunner
motor similar to the ODrive D6374. [49]
illustrates the interior construction of a typical hobby-grade outrunner motor. Of
particular note are the concentrated stator windings, 12 wire-wound stator poles,
and 14 permanent magnet poles. This is also considered an outrunner motor since
the rotor is outside the stator. These properties are the same between the KDE
Direct KDE2315XF-885 and the ODrive D6374, which makes this illustration a
good representation of the interior construction of the ODrive 6374.
The ODrive D6374 hobby motor is sold as a brushless direct current motor
(BLDC). Despite its name, it is actually a permanent magnet synchronous motor
97
(PMSM). This is because the back-EMF is a sinusoidal waveform, not a trape-
zoidal waveform. The ODrive 6374 motor can be modeled using the three-phase
circuit diagram given in Fig. 5.4. This diagram illustrates the phase resistance,
phase inductance, and back-EMF, along with the phase voltage inputs. However,
as derived in Section 5.2, the ODrive D6374 motor can be analyzed using the DQZ
transformation to simplify many of the computations.
ec
ea
eb
La
Lb
Lc
Ra
Rb
Rc
Va
Vb
Vc
n
Ia
Ib
Ic
Figure 5.4: Circuit diagram of a three-phase PMSM, such as the ODrive D6374.
In order to analyze the ODrive motor, the phase resistance and phase in-
ductance must be identified. Further, the torque and back-EMF must also be
quantified for a particular input current. This requires identification of the speed
and torque constants. Several motor properties of the D6374 are provided by
ODrive and are given in Table 5.2.
ODrive D6374 Motor Properties
KV (
rpm
VLL,pk
) Max Voltage
(VLL,pk)
Max Current
(Apk)
Phase
Resistance
(mΩ)
Stator
Poles/Magnet
Poles
150 48 70 39 12n14p
Table 5.2: Motor properties for the ODrive D6374 [36].
98
Although not explicitly given by ODrive, the torque constant can be derived
from the speed constant. The equation for the motor torque constant is given in
Eq. (5.1). Thus, the torque constant is found to be 0.0551.
The phase resistance and the phase inductance are two additional motor pa-
rameters that are important for modeling the dynamics of the motor. While
the phase resistance is given by ODrive, the phase inductance is not. However,
both of these parameters can be measured during start-up calibration. For the
ODrive D6374 used during this chapter, both of these parameters are measured
and are given in Table 5.3. The torque constant is also provided in this table. In
Inferred/Measured ODrive D6374 Motor Properties
KT (
Nm
Apk
) Phase Resistance (mΩ) Phase Inductance (µH)
0.0551 44 23.77
Table 5.3: Inferred and measured motor properties for the ODrive D6374.
many cases, the phase inductance is measured phase-phase. However, according
to ODrive documentation, the phase inductance is measured phase-neutral [50].
Thus, the d-axis inductance and q-axis inductance are equal to the phase induc-
tance.
The motor moment of inertia is calculated as the sum of the rotor moment of
inertia and the winch moment of inertia. The rotor moment of inertia is provided
by ODrive as approximately 1 × 10−4 kg-m2 [36]. The winch moment of inertia
is calculated using the mass properties tool in Solidworks and the total filament
used as predicted by the Prusa Slicer software. Using the Prusa Slicer software,
the winch is found to be 12.91 g. Using this mass and the model for the winch in
Solidworks, the moment of inertia for the winch is found to be 0.66164 kg-m2.
The last remaining parameters of the ODrive D6374 are the frictional coeffi-
cients. The ODrive is modeled using a combined Coulomb-viscous-static friction
model. This is because the torque required to start rotating from rest is observed
to be higher than the torque required to maintain a constant velocity.
The friction parameters are measured using two different tests. The first test
99
measures the torque required for the motor to start rotating from rest. For this
test, a known mass is attached to a cable and held taut but not pulling. The motor
is set to torque mode, and the torque is incremented until the motor is able to
lift the mass. Three different trials are conducted using a different mass for each
trial. The second test measures the acceleration of the motor when the motor is
moving. As before, a known mass is attached to the cable and held taut but not
pulling. The motor is again set to torque mode and set to an arbitrary torque set
point. The position, velocity, and applied torque are recorded for time steps of
0.05 s. The acceleration is calculated by differentiation. Finally, the transience is
removed from the initial motor motion to ensure the motion has reached steady
state. Using this acceleration, torque, and the known load mass, the total friction
for each time step can be found by plugging into Eq. (5.20). In this test, six trials
are conducted. Finally, least squares analysis is used to determine the frictional
coefficients.
The results for each of the static friction trials are given in Table 5.4. In
particular, the masses, calculated load torque, torque required to start rotation,
and calculated static friction for each trial are provided.
Trial 1 Trial 2 Trial 3
Mass (kg) 0.20025 0.99881 2.7178
Load
Torque (Nm)
0.0125 0.0622 0.1693
Minimum
Rotation
Torque (Nm)
0.09 0.135 0.225
Calculated
Static Friction
(Nm)
0.0775 0.0728 0.0557
Table 5.4: Experimental parameters and derived coefficients for the static friction
evaluation using the ODrive D6374.
100
The average static friction is calculated to be 0.0687 Nm. These results are
shown to have somewhat significant variance across the three trials. This is as-
sumed to be caused by cogging torque, which is defined as the attraction between
the stator teeth and the permanent magnets in a PMSM. This cogging torque
varies based on the alignment of the rotor, and as such is likely the cause of the
variation in the static friction results. This cogging torque cannot be reduced as
it is a property of the motor. However, it can be compensated by the implementa-
tion of an anticogging algorithm. In the case of this static friction test, the effect
of the cogging torque on the results can be mitigated by increasing the number of
trials. This would provide the average static friction based on the average cogging
torque in the motor.
The parameters chosen for each of the kinetic friction trials are given in Ta-
ble 5.5. In addition, the total friction at each time step is shown in Fig. 5.5. Using
linear least-squares regression, the line of the best is recorded and plotted on this
data. This illustrates the coefficients that correspond to Bv and Bc in Eq. (5.20).
The coefficients of the line of best fit are recorded in Table 5.6.
As in the static friction results, these results are shown to have somewhat
significant variance across the three trials. This is assumed to be caused by cogging
torque. In order to determine the effect of cogging torque on the system, the least-
squares regression could be modified to include the cogging torque as an additional
parameter. In addition, the position of the rotor would have to be considered for
each time step. If this variance is indeed caused by cogging torque, one would
expect the total friction torque to be the same at fixed rotation increments.
101
Trial 1 Trial 2 Trial 3 Trial 4 Trial 5 Trial 6
Mass (kg) 1.35886 1.35886 2.27955 2.27955 2.7178 2.7178
Load Torque
(Nm)
0.0846 0.0846 0.1420 0.1420 0.1693 0.1693
Motor Torque
(Nm)
0.15 0.175 0.21 0.235 0.235 0.25
Table 5.5: Experimental parameters for the kinetic friction evaluation using the
ODrive D6374.
Figure 5.5: Total friction vs. motor velocity for the ODrive D6374.
ODrive D6374 Kinetic Friction Coefficients
Bv (Nm-s) Bc (Nm)
1.4152× 10−4 0.0598
Table 5.6: Kinetic friction coefficients for the ODrive D6374.
102
5.4 Controller Design for the ODrive D6374
The controller for the ODrive D6374 is implemented by ODrive as a cascaded
position, velocity, and current controller [51]. The overall control structure is
shown in Fig. 5.6.
Figure 5.6: Cascaded position, velocity, and current controller implemented by
ODrive [51].
Each of the stages is a PI controller with the exception of the position control
loop, which is a position controller. The proportional and integral gains of each
stage are tuned independently to achieve the desired bandwidth and performance
specifications. Note that the feedforward term is set to zero in all cases by default.
The input expression for each control stage is given by ODrive as
ωm,r[k] = K
d
p,p(pr[k]− p[k]), (5.23)
Iω[k] = Iω[k − 1] +Kdi,ω(ωm,r[k]− ωm[k]), (5.24)
Iq,r[k] = K
d
p,ω(ωm,r[k]− ωm[k]) + Iω[k], (5.25)
Vi,a[k] = Vi[k − 1] +Kdi,I(Ia,r[k]− Ia[k]), (5.26)
Va,r[k] = K
d
p,I(Ia,r[k]− Ia[k]) + Vi,a[k]. (5.27)
The definitions for each of the parameters in Eqs. (5.23)-(5.27) are given in Ta-
ble 5.7. Note that these controllers are described in discrete time, with sampling
103
time Ts defined as 125 µs. However, the motor equations described in Eqs. (5.14)-
(5.22) are in continuous time.
Symbol Description
pr Reference motor position (rad)
p Feedback motor position (rad)
Kdp,p Discrete proportional gain with respect
to position error
ωm,r Reference motor velocity (
rad
s
)
ωm Feedback motor velocity (
rad
s
)
Kdi,ω Discrete integral gain with respect
to velocity error
Iω Integral state with respect to velocity
error
Kdp,ω Discrete proportional gain with respect
to velocity error
Ia,r Reference a-axis motor current (A),
where a is either q or d
Ia Feedback a-axis motor current (A),
where a is either q or d
Kdi,I Discrete integral gain with respect to
current error
Vi Integral state with respect to current
error
Kdp,I Discrete proportional gain with respect
to current error
Va,r Reference a-axis motor voltage (V)
Table 5.7: Parameters of the ODrive D6374 controller given in Eqs. (5.23)-(5.27).
The implementation of this discrete time controller is somewhat atypical com-
pared to a traditional discrete time controller. This controller can be derived
from the continuous time equivalent controller, which is necessary for determining
the discrete time gains. In addition, the simulations in the following section are
conducted in continuous time, so this transformation is necessary for determining
the continuous time controller.
104
The general continuous time PI controller is defined as
u(t) = Kp(r(t)− y(t)) +Ki
∫ t
0
(r(τ)− y(τ))dτ,
= Kp(r(t)− y(t)) + I(tk−1) +Ki
∫ t
tk−1
(r(τ)− y(τ))dτ,
(5.28)
where,
I(t) = Ki
∫ t
0
(r(τ)− y(τ))dτ,
=
∫ t
0
f(τ)dτ.
(5.29)
The transformation from the continuous time to the discrete time controller is
straightforward. The proportional controller is translated directly to the discrete
domain, with the continuous time t corresponding to the discrete step Tsk. The
discrete integral controller is translated from the continuous time using a Riemann
sum. This is clear from the definition of the Riemann sum. This is given as
F [k] = ∆Ts
k∑
i=0
f(ti),
= F [k − 1] + ∆Tsf(tk).
(5.30)
This equation matches the form of Eqs. (5.24) and (5.26). Here, Kdi,ω(ωm,r[k] −
ωm[k]) = ∆Tsf(tk) and K
d
i,I(Ir[k]− I[k]) = ∆Tsf(tk) in Eq. (5.24) and Eq. (5.26),
respectively. Thus, it is clear that the continuous integral gain can be translated
to discrete time using the equation
Ki =
Kdi
∆Ts
. (5.31)
These steps have shown how the continuous gains can be translated to discrete
time. However, based on analysis, it appears the gains provided in the ODrive
configuration are already in continuous time. This is shown in the derivation of
105
the proportional and integral gains for the current controller. The rest of this
chapter assumes the gains in the ODrive configuration are already in continuous
time. If this is shown to be incorrect, the simulation can be modified using the
equations derived above.
The ODrive current controller determines the output current given a refer-
ence current. The proportional and integral gains are determined according to
a critically damped controller with a user-defined 3-dB bandwidth. The default
bandwidth is 1000 rad
s
, which is used to derive the proportional and integral gain
parameters. Using Eqs. (5.14) and (5.15), the current loop can be defined for both
the d-axis and the q-axis. Finally, this derivation is found assuming ωe = 0, which
is standard when designing a motor controller. The control loop for the d-axis
and the q-axis are described in continuous time as
Vd(t) = Id(t)Rs + Ld
dId(t)
dt
= Kp,d(Ir,d(t)− Id(t)) +Ki,d
∫ t
0
(Ir,d(τ)− Id(τ))dτ,
(5.32)
Vq(t) = Iq(t)Rs + Lq
dIq(t)
dt
= Kp,q(Iq,r(t)− Iq(t)) +Ki,q
∫ t
0
(Iq,r(τ)− Iq(τ))dτ.
(5.33)
Notably, the differential equation is the same for both the d-axis and the q-axis
current. The transfer function for the d-axis and q-axis currents are thus derived
as
Ia(s)
Ia,r(s)
=
Kp,as+Ki,a
Las2 + (Kp,a +Rs)s+Ki,a
, (5.34)
where a denotes the current axis, either d or q. The 3-dB frequency can be found
by expressing the squared magnitude of Eq. (5.34) and setting this equal to 1
2
.
This is given as∣∣∣∣ Ia(jωc)Ia,r(jωc)
∣∣∣∣2 = 12 = (Kp,aωc)2 +K2i,a(Ki,a − Laω2c )2 + (Kp,aωc +Rsωc)2 , (5.35)
106
where the damping ratio and natural frequency are defined as
ζ = 1 =
Kp,a +Rs
2Laωn
, (5.36)
ωn =
√
Ki,a
La
. (5.37)
Using Eqs. (5.35)-(5.37), the proportional and integral gains for the current loop
are found to be Kp,a = 0.019672
V
A
and Ki,a = 42.639167
V
A-s
. These gains match
the values given in the ODrive configuration for 1000 rad
s
. Since these gains are
derived in continuous time, this implies that the gain parameters in the ODrive
configuration are also in continuous time. Therefore, the gain parameters defined
in the ODrive configuration for the velocity and position feedback loops are also
assumed to be in continuous time.
In the simulation, the current feedback differential equation is differentiated
with respect to t. This is found to have better simulation result stability. The
resulting differential equation is given as
dId(t)
dt
Rs + Ld
d2Id(t)
dt2
− ωe(t)Lq dIq(t)
dt
− dωe(t)
dt
LqIq(t)
= Kp,d(
dIr,d(t)
dt
− dId(t)
dt
) +Ki(Ir,d(t)− Id(t)),
(5.38)
dIq(t)
dt
Rs + Lq
d2Iq(t)
dt2
+ ωe(t)Ld
dId(t)
dt
+
dωe(t)
dt
LdId(t) +
dωe(t)
dt
λpm
= Kp,q(
dIq,r(t)
dt
− dIq(t)
dt
) +Ki(Iq,r(t)− Iq(t)).
(5.39)
The velocity control loop is not defined explicitly, but the output current signal is
defined with respect to the reference velocity. This is defined in continuous time
as
Iq,r(t) = Kp,ω(ωm,r(t)− ωm(t)) +Ki,ω
∫ t
0
(ωm,r(τ)− ωm(τ))dτ. (5.40)
The position control loop is also defined similarly. Unlike the current and velocity
107
control loops, the position control loop is a P controller. This is defined as
ωm,r(t) = Kp,p(pr(t)− p(t)). (5.41)
The proportional and integral velocity gains in this equation are defined in the
ODrive configuration. Note that the position and velocity in the ODrive config-
uration are defined in Nm as well as rotations, whereas these gains are derived
assuming units of A and radians. Using the tuning configuration guide given
by [51], these gains are set to Kp,p = 35
rad/s
rads
, Kp,ω = 1.1547
A
rad/s
, and Ki,ω =
11.5470 A
rad/s-s
. In the ODrive configuration, these correspond to Kp,p = 35
rev/s
rev
,
Kp,ω = 0.4
Nm
rev/s
, and Ki,ω = 4
Nm
rev/s-s
.
5.5 Simulation Results
The equations derived in Sections 5.3 and 5.4 are used to simulate the dynamics of
the motor in the CDCM. This is simulated according to the Solar Cruiser model,
which has a 29 m TRAC boom and material properties derived in Chapter 2.
For this simulation, the load torque is modeled using a 3rd-order polynomial in
a similar manner to the forward kinematics analysis in Chapter 3. This polyno-
mial describes the load torque with respect to the change in motor position. This
polynomial is derived using the required cable tension bounds derived in Chap-
ter 2, which was found to be (0 N, 85 N). The polynomial is generated using 341
discrete points generated using the Chebyshev nodes described in Section 3.2.2.
This polynomial function is found to be
Tl = 0.0037p
3 − 0.0414p2 + 0.2985p− 1.4047× 10−4. (5.42)
The first simulation actuates the motor from a cable tension of 0 N to the
maximum 85 N. This assumes the cable is taut but not pulling at its initial
position. This simulation further assumes an ideal motor in open-loop velocity
control, i.e., the inductance, rotor moment of inertia, and friction are all zero.
108
Therefore, the d-axis current and q-axis current can be controlled directly by the
voltage set point. In order to achieve the maximum 85 N cable tension set point,
the voltage is related to the current, which is related to the motor torque, which
is in turn related to the cable tension. This is given as
Vq = IqRs + ωeλpm =
Te
KT
Rs +
ωm√
3KV,rad
, (5.43)
Te = Tl = KT Iq, (5.44)
Tension (N) =
Torque (Nm)
radius (rad)
. (5.45)
Noting the winch radius of 0.25 in, the required torque is found to be 0.5397 Nm.
Similarly, the required q-axis current is found to be 9.7900 A. Since there is no
motor velocity when the CDCM has reached equilibrium, the q-axis voltage is
equivalent to the voltage drop across the phase resistance. Thus, the q-axis voltage
is found by determining the voltage across the resistor at equilibrium, which is
found to be 0.4308 V. Under these assumptions, the position and velocity of the
rotor are plotted. The current and total power is also plotted, where total power
is defined as the sum of the mechanical and electrical power. These are shown in
Figs. 5.7 and 5.8.
While these results provide a good baseline performance, they are not very
accurate with respect to the non-idealities of the motor. Therefore, the simulation
is modified to simulate closed-loop feedback control of the motor in torque control.
This simulation includes the motor parameters derived in Section 5.3. The current
set-point is determined from the maximum cable tension of 85 N, which is found
to be 9.7900 A. This is the same as the results in the previous simulation since
the positional equilibrium is the same in both cases. Using Eqs. (5.14)-(5.22),
and the current feedback equations derived in Eqs. (5.38) and (5.39), the motor
is simulated in torque control. These results are shown in Figs. 5.9 and 5.10.
As this simulation implements all of the non-ideal model parameters of the
D6374 motor, these results should more accurately reflect the motor motion. As
109
(a) (b)
Figure 5.7: Simulation results for the CDCM with the TRAC boom assuming
constant voltage. This includes (a) the position and (b) the velocity of the motor
with respect to time.
(a) (b)
Figure 5.8: Power requirement simulation results for the CDCM with the TRAC
boom assuming constant voltage. This includes (a) the mechanical power and (b)
the electrical power losses of the motor with respect to time.
110
(a) (b)
Figure 5.9: Simulation results for the CDCM with the TRAC boom using torque
control. This includes (a) the position and (b) the velocity of the motor with
respect to time.
(a) (b)
Figure 5.10: Power requirement simulation results for the CDCM with the TRAC
boom using torque control. This includes (a) the mechanical power and (b) the
electrical power losses of the motor with respect to time.
111
such, it is clear that torque control is insufficient for accurately controlling the
motor position. As shown by Fig. 5.9a, the equilibrium position has some steady-
state error. This is caused by the friction coefficients, where the motor torque is
at the desired set point, but the friction causes the useful mechanical torque to
be reduced. As a result, the motor is offset from the desired equilibrium position.
In theory, this steady-state error in the net torque can be as large as Bs,max, since
the motor is unable to overcome this friction to start moving.
In order to overcome the limitations of torque control, the cascaded position
control loop is implemented in simulation. In this case, the speed set point and
the current set point are both controlled by the cascade feedback loop. Thus, only
the position set point needs to be determined. The position set point is found by
running the PRB simulation with the TRAC boom material properties using a
85 N cable tension [20, 23]. The change in cable length is recorded and translated
from units of meters to radians using the known winch radius. Note that the sign
of the change in cable length is flipped to ensure a positive change in cable length
corresponds to a positive beam deflection. The resulting motor position set point
is found to be 2.4711 rads.
Rather than setting the desired position directly, ODrive has developed a fil-
tered position function that ensures the motor follows a smooth trajectory while in
position control. This is because the bandwidth of the position filter is quite large,
so the response of the unfiltered position control would result in fast, jerky motion.
The filtered position control is implemented in simulation as well. The filtered
position control is designed to emulate a critically damped mass-spring-damper
with a user-defined bandwidth. The mass-spring-damper can be expressed as
Fext = m
d2x(t)
dt2
+ c
dx(t)
dt
+ kx(t). (5.46)
The mass-spring-damper is tuned to have a bandwidth of 2 rad
s
. However, the
definition of bandwidth in this context is ambiguous. In the current feedback
controller, the bandwidth is defined as the 3-dB roll-off frequency of the transfer
112
function. This is confirmed by the proportional and integral gains in the ODrive
configuration matching the calculated values. However, there are no proportional
and integral gains in the ODrive configuration for the filtered position filter. Thus,
the proportional and integral gains are derived using two different methods of
bandwidth. The first is the 3-dB bandwidth as in the current feedback controller.
The second is the natural frequency.
Assuming a desired position pr(t), the tunable mass-spring-damper equation
can be defined as
0 =
d2pr(t)
dt2
− d
2p(t)
dt2
+Kp,r(
dpr(t)
dt
− dp(t)
dt
) +Ki,r(pr(t)− p(t)). (5.47)
It is assumed that both d
2pr(t)
dt2
and dpr(t)
dt
are zero. This is because the set point of
a mass-spring-damper would not change. Therefore, the transfer function for this
system can be derived as
p(s)
pr(s)
=
Ki,r
s2 +Kp,rs+Ki,r
. (5.48)
The proportional and integral gains of this filtered position system are first
derived using the same method as was used for the current controller gains. These
gains correspond to a 3-dB magnitude at 2 rad
s
. The system of equations is defined
explicitly as ∣∣∣∣ p(s)pr(s)
∣∣∣∣2 = 12 = K2i,r(Ki,r − ω2c )2 + (Kp,rωc)2 , (5.49)
where the damping ratio and natural frequency are defined as
ζ = 1 =
Kp,r
2ωn
, (5.50)
ωn =
√
Ki,r. (5.51)
The resulting proportional and integral gains are found to be Kp,r = 6.2151 and
Ki,r = 9.6569.
113
The proportional and integral gains of this filtered position system are also
derived with a 2 rad
s
natural frequency. The transfer function is defined in this
case as
p(s)
pr(s)
=
Ki,r
s2 +Kp,rs+Ki,r
=
Ki,r
s2 + 2ζωns+ ω2n
. (5.52)
For ζ = 1 and ωn = 2, the proportional and integral gains are found to be Kp,r = 4
and Ki,r = 4.
In the following section, it is shown that the filtered position function that
uses the bandwidth definition corresponding to the natural frequency has more
accurate results than the bandwidth definition corresponding to the 3-dB fre-
quency. Therefore, the simulations in the remainder of this section are run using
the filtered position function defined in Eq. (5.52).
Using the equations defined in Eqs. (5.14)-(5.22), along with the controller
feedback equations derived in Eqs. (5.38)-(5.41), and the filtered position function
in Eq. (5.52), the motor is simulated in position control. The results of this
position control simulation are shown in Figs. 5.11 and 5.12.
(a) (b)
Figure 5.11: Simulation results for the CDCM with the TRAC boom using posi-
tion control. This includes (a) the position and (b) the velocity of the motor with
respect to time.
114
(a) (b)
Figure 5.12: Power requirement simulation results for the CDCM with the TRAC
boom using position control. This includes (a) the mechanical power and (b) the
electrical power losses of the motor with respect to time.
These results confirm the feasibility of the position controller for the ODrive
D6374 motor. Despite the non-idealities in the motor, the controller is able to
effectively track the desired position with zero steady-state error. It is also able to
do so while implementing a relatively smooth trajectory due to the filtered position
function. Based on these results, it is determined that a position controller is the
optimal system for the CABLESSail CDCM controller.
5.6 Experimental Results
The dynamic motor model is compared to the experimental results generated
during testing of the polycarbonate beam CDCM in Section 4.4. This experiment
applied a filtered position command to the ODrive, with position set points from
0.1 rotations to 0.8 rotations in 0.1 rotation increments. For each trial, the motor
position, motor velocity, and motor torque are recorded over a 15 s interval. The
equilibrium end-effector deflection values are recorded by taking the average value
115
between the time interval 12 s and 14 s for both the simulated and experimental
results. The simulation results are compared to experimental results for motor
position set points of 0.2, 0.4, and 0.6 rotations (1.2566, 2.5133, and 3.7699 rads).
The load torque model is modified to reflect the CDCM model developed in
Section 4.4. The PRB model is used to generate the end-effector deflection and
the change in cable length between sample tension forces [20]. The necessary
sample tension force bounds were shown to be in the range of (0 N, 200 N) in
Section 4.4. A total of 801 data points are generated according to the Chebyshev
nodes, and a 6th-order polynomial is fitted to each of the desired functions. The
first function is the load torque vs. change in motor position, similar to the one
described in Eq. (5.42). The second function relates the change in the end-effector
position to the load torque. Since the first function derives a relationship between
load torque and the change in motor position, This means these functions can be
composed to define a function that relates the end-effector position to the change
in motor position. These two functions are found to be
Tl = −1.7857× 10−7p6 + 6.5142× 10−6p5 − 1.1776× 10−4p4
+0.0016p3 − 0.0195p2 + 0.2630p+ 1.6679× 10−6,
(5.53)
x1 = 0.0779T
6
l − 0.1305T 5l − 0.0550T 4l − 0.2694T 3l
+0.1710T 2l + 0.8688Tl + 6.1969× 10−6.
(5.54)
These functions are identical to the functions found in Table 4.4.
Using the motor model defined in Eqs. (5.14)-(5.22), the controller feedback
equations derived in Eqs. (5.38)-(5.41), the load torque equation in Eq. (5.53),
and the filtered position function in Eq. (5.52), the motor simulation is run in
position control. The simulated load torque at each time t is used to deter-
mine the simulated CDCM end-effector deflection. These results are shown in
Figs. 5.13, 5.14, and 5.15.
These results compare similarly to the results found in Section 4.4. Specifically,
the motor is able to effectively track the desired motor position, but the torque and
116
(a) (b)
(c)
Figure 5.13: Simulation and experimental motor position results for the CDCM
with the polycarbonate beam using position control. Shown are the position set
points of (a) 1.2566 rads, (b) 2.5133 rads, and (c) 3.7699 rads.
117
(a) (b)
(c)
Figure 5.14: Simulation and experimental motor torque results for the CDCM
with the polycarbonate beam using position control. Shown are the position set
points of (a) 1.2566 rads, (b) 2.5133 rads, and (c) 3.7699 rads.
118
(a) (b)
(c)
Figure 5.15: Simulation and experimental end-effector deflection results for the
CDCM with the polycarbonate beam using position control. Shown are the posi-
tion set points of (a) 1.2566 rads, (b) 2.5133 rads, and (c) 3.7699 rads.
119
the end-effector deflection are greatly overestimated. These results indicate some
amount of cable stretching. The disparity between the simulated and experimental
end-effector deflection is shown in Table 5.8.
Motor
Set Point (rads)
Simulated
Deflection (mm)
Experimental
Deflection (mm)
1.2566 270.3179 45.17
2.5133 481.6845 162.17
3.7699 613.0873 323.97
Table 5.8: Simulated and experimental deflection of the end-effector with the
polycarbonate beam.
As the magnitude of displacement increases, the ratio of the experimental
deflection to the simulated deflection also increases. This makes sense, as the
change in cable stretch is likely greatest for lower cable tensions than higher
tensions. Thus, as the CDCM moves from the zero set point to 0.2 rotations,
the effect of the cable stretch is likely very high. In contrast, as the CDCM moves
from 0.4 rotations to 0.6 rotations, the effect of the cable stretch is likely much
smaller.
In order to improve the simulation to more accurately model the dynamics
of the CDCM, this cable stretching must be included in the model. A simple
way to do this is to assume the cable stretching ratio is constant. Although it
has been shown in Table 5.8 that this is not true, this is useful as a qualitative
analysis to determine if the model can be significantly improved by simply adding
the effect of cable stretching. For each of the motor set points discussed in Ta-
ble 5.8, the simulated motor position that results in the experimental end-effector
displacement is found. This is found by identifying the simulated end-effector
position with the closest correspondence to the experimental end-effector position
from the data generated in Section 5.6. Then, the motor position correspond-
ing to this index is recorded. This method is used instead of the polynomial fit
function because the motor position with respect to the end-effector position is
not unique, as very large cable tensions (and, correspondingly, changes in motor
120
position) cause the magnitude of the end-effector displacement to change from
increasing to decreasing. Since the motor position is not unique with respect to
the end-effector displacement, this cannot be represented as a function. In the
case where two different motor positions corresponded to the same end-effector
displacement, the corresponding motor position that is less than the motor po-
sition set point is selected. The recorded motor positions corresponding to the
experimental end-effector displacements are shown in Table 5.9.
Motor
Set Point (rads)
Experimental
Deflection (mm)
Corresponding
Motor
Position (rads)
Position
Ratio
1.2566 45.17 0.1991 0.1584
2.5133 162.17 0.7332 0.2917
3.7699 323.97 1.5345 0.4070
Table 5.9: Motor position corresponding to the experimental end-effector deflec-
tion and the resulting position ratio between the corresponding motor position
and the motor set point.
These corresponding motor positions are lower than the true set points since
the model assumes no cable stretching. Intuitively, this position ratio represents
the change in position of the cables compared to the change in position of the
motor. The modified position expression for the change in cable position compared
to the change in motor position is given as
pstretch(t) = ηprp(t), (5.55)
where ηpr is the position ratio for the particular motor position set point. This
modified cable position is used in the simulation, which propagates to the deter-
mination of the load torque and the end-effector deflection.
The results of this modified simulation are compared to the experimental re-
sults for 0.2, 0.4, and 0.6 rotations (1.2566, 2.5133, and 3.7699 rads). These result
are shown in Figs. 5.16, 5.17, and 5.18.
These results show significant improvement compared to the previous results.
121
(a) (b)
(c)
Figure 5.16: Modified simulation and experimental motor position results for
the CDCM with the polycarbonate beam using position control. Shown are the
position set points of (a) 1.2566 rads, (b) 2.5133 rads, and (c) 3.7699 rads.
122
(a) (b)
(c)
Figure 5.17: Modified simulation and experimental motor torque results for the
CDCM with the polycarbonate beam using position control. Shown are the posi-
tion set points of (a) 1.2566 rads, (b) 2.5133 rads, and (c) 3.7699 rads.
123
(a) (b)
(c)
Figure 5.18: Modified simulation and experimental end-effector deflection results
for the CDCM with the polycarbonate beam using position control. Shown are
the position set points of (a) 1.2566 rads, (b) 2.5133 rads, and (c) 3.7699 rads.
124
Using a linear approximation between the input position set point and the motor
position corresponding to the experimental end-effector position, the simulation
is able to better match the end-effector position and motor torque. However, the
position ratio is calculated uniquely for each position set point. The position ratio
also varied greatly between trials. Thus, while it confirms that the motor model
can effectively predict these parameters if it is given an accurate load torque,
it does not provide an expression for the load torque that is generalizable when
there is a nominal cable stretch. Further work would be needed to more accurately
model this cable stretch, or otherwise replace the cables such that the stretch is
negligible.
The experimental q-axis current is derived using Eq. (5.19). Using this current,
the simulated and experimental power requirements are plotted for the set points
of 0.2, 0.4, and 0.6 rotations (1.2566, 2.5133, and 3.7699 rads). These are shown
in Figs. 5.19 and 5.20.
It is noted that the experimental loss power results are underestimated com-
pared to the true loss power. While the q-axis current can be inferred from the
motor torque, the d-axis current was not recorded during the experiments. As a
result, the loss power is calculated using only the q-axis current and assuming the
d-axis current is zero. According to simulation, this d-axis current never exceeded
0.5 mA of current at any point. Thus, the assumption that this d-axis current is
zero appears to be valid. However, it is important to note this limitation and to
consider determining the magnitude of the d-axis current in hardware.
These results show that the simulation can predict the power requirements
with reasonable accuracy. Notably, all of these results have more accurate results
for higher cable tensions and beam deformations. Some of this is likely caused
by cable stretching; however, there are other sources of potential error, such as a
non-uniform beam or a nonzero initial beam deformation. It is also noted that
the PRB model seems to have a bug in the deformation since it does not seem to
consider the effects of gravity.
125
(a) (b)
(c)
Figure 5.19: Mechanical power requirement simulation and experimental results
for the CDCM with the polycarbonate beam. Shown are the position set points
of (a) 1.2566 rads, (b) 2.5133 rads, and (c) 3.7699 rads.
126
(a) (b)
(c)
Figure 5.20: Electrical power losses simulation and experimental results for the
CDCM with the polycarbonate beam. Shown are the position set points of (a)
1.2566 rads, (b) 2.5133 rads, and (c) 3.7699 rads.
127
5.7 Closing Remarks
This chapter combined many of the concepts developed in the previous chapters
to derive a model of the CDCM from the frame of reference of the motor. These
results evaluated the accuracy of the PRB model used in many of the other chap-
ters, demonstrated its limitations, and provided a method to improve its accuracy
compared to a real CDCM. The results of this chapter also detail the steps to
derive the power requirements of the prototype CABLESSail model, while also
providing a quantitative analysis of these requirements.
While the motor model was derived from theory, several limitations of this
model exist and should be noted. The most significant approximation was likely
the load torque model. As shown in this chapter, the cable stretch has a significant
effect on the CDCM end-effector displacement as well as the load torque in the
motor. However, it was also noted that this model was derived assuming a static
equilibrium. As a result, the dynamics of the CDCM were not considered in this
model. It is also noted that the CDCM likely introduces additional damping and
inertia into the system that was not modeled in this simulation. As future work,
the results of this chapter indicate that the beam and cable models will be the
most important areas of improvement for this dynamic model.
Overall, this chapter provided an in-depth analysis of the ODrive D6374 motor
for the CABLESSail project. It also provided the theory and discussion of motor
parameters as they pertain to the operation of the motor. As a result, this chapter
can be used to evaluate the feasibility of PMSMs beyond the ODrive D6374 for
future use in this project.
Chapter 6
Conclusion
The concept of the solar sail promises to push the boundaries of both spacecraft
and space exploration. Current space missions are limited in scope largely by
the cost and mass of fuel required for propulsion. Solar sails have none of these
limitations, providing a lightweight structure and a theoretically infinite supply
of propulsion. This gives them incredible flexibility in their range of feasibility.
However, the design of the solar sail has several trade-offs. First, its lightweight
construction limits its structural integrity. Second, its large sail area causes it to
have a large inertia. Finally, the solar sail requires a method for attitude control as
well as stabilization. These challenges must be overcome in order for the concept
of the solar sail to become a viable spacecraft.
This thesis detailed the work on the CABLESSail project, a novel solar sail
concept that uses cable-driven actuation to control flexible booms in order to
achieve stability and attitude control. Chapter 2 introduced the theory of the
solar sail as well as the concept of the CDCM. This chapter also provided design
criteria based on NASA specifications. Chapter 3 discussed some of the methods
of estimating the end-effector pose using the known cable length or cable ten-
sion. Chapter 4 reviewed the hardware design and implementation of the CDCM.
It also provided experimental results for this work, which were compared to the
simulations developed in the previous chapters. Finally, Chapter 5 developed a
128
129
high-fidelity motor model to model the dynamics of the cable robot. It also as-
sessed the actuation and power requirements of the cable robot and provided both
experimental and simulation results of these parameters. The analysis and hard-
ware development discussed in this thesis lay the groundwork for the continued
development of a full-scale CABLESSail prototype.
This thesis work also leaves open several important aspects for the continuation
of the CABLESSail project. Many of the boom and sail properties were inferred
using the published works of existing solar sail projects. However, the particular
boom and sail designs for the Solar Cruiser and Solar Polar Imager models are
unclear. While this thesis provided the equations and the outline for a solar
sail analysis using known properties, these properties must be updated to reflect
the Solar Cruiser and Solar Polar Imager models once these are known. Much
work is also needed to accurately model the statics and dynamics of the beam.
This thesis assumed a static equilibrium relationship between the end-effector
pose and the cable tensions and lengths. However, this could lead to excessive
stress on the beam if the bounds of safe operation are dynamically exceeded.
More significantly, the static model was shown to have some modeling error likely
caused by cable stretching. In future work, it would be important to consider
an estimation method for the end-effector pose. The most likely candidate is a
Kalman Filter with sensor feedback.
Overall, this thesis has aimed to provide much of the necessary background
on the theory of both the solar sail and the CDCM for use with the CABLESSail
prototype. This thesis has provided a generalized analysis of the many aspects of
this work for ease of integration with future designs, while also providing quanti-
tative results for the particular hardware used in the first three prototypes. Most
significantly, these results show promise for the feasibility of the CABLESSail con-
cept. With continued work, this may enable the viability of the solar sail concept
for future spaceflight.
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135
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Appendix A
Solar Sail Beam Deformation for
the Collapsible Tubular Mast
Beam
The boom is modeled according to the CTM design given by Fig. 2.5. The tubular
cross-section is approximated as a hollow disk with an inner diameter of 9.57 mm
and an outer diameter of 9.85 mm. It is assumed that the deformation will occur
in the longitudinal direction, thus the elastic modulus of E11, Poisson’s ratio of
v12, and the shear modulus of G12 will be used to determine the magnitude of
deformation, given in Table 2.4.
Figs. A.1 and A.2 illustrate the beam deformation required for a CTM beam
using these material and cross-section properties. Each of these plots assumes 20
distinct beam element disks for cable routing.
136
137
(a) (b)
(c)
Figure A.1: Pitch/Yaw torque generated from lateral boom deformation for Solar
Cruiser with a 2 cm cable radius and 8.2 N of maximum tension. The plots include
the torque generated at (a) 0° SIA, (b) 35° SIA normal to the axis of deformation,
and (c) 35° SIA along the axis of deformation.
138
(a) (b)
(c)
Figure A.2: Pitch/Yaw torque generated from lateral boom deformation for Solar
Polar Imager with a 2 cm cable radius and 6.45 N of maximum tension. The
plots include the torque generated at (a) 0° SIA, (b) 35° SIA normal to the axis
of deformation, and (c) 35° SIA along the axis of deformation.
Appendix B
Additional Cable-Driven
Continuum Manipulator Figures
139
140
B.1 Iteration I
Figure B.1: Engineering Diagram of the first iteration of the CDCM base. All
units are in mm.
141
Figure B.2: Isometric model of the first iteration of the CDCM base with the
motor enclosure removed.
142
B.2 Iteration III
Figure B.3: Engineering Diagram of the third iteration of the CDCM base. All
units are in mm.
143
Figure B.4: Isometric model of the third iteration of the CDCM base with the
motor enclosures removed.
Figure B.5: Isometric model of the interior of the third iteration of the CDCM
base.
144
Figure B.6: Isometric model of the tape measure TRAC boom with attached
beam elements.
Appendix C
Prototype Cable-Driven
Continuum Manipulator Parts
List
Item Seller Part
Number
Purpose Quantity
Filament Amazon 3D PLA-
1KG1.75-
BLK
3-D printed parts 5
T-Slotted
Framing
McMaster-
Carr
47065T101 CDCM frame 1
Ball Bearing McMaster-
Carr
60355K505 Support spool
and reduce
friction
4
T-Slotted
Fastener
McMaster-
Carr
47065T142 Attach
components to
T-Slotted
Framing
9
145
146
T-Slotted
Fastener
(Button
Head)
McMaster-
Carr
47065T139 Attach motor
enclosure plate to
T-Slotted
Framing (smaller
head for
clearance)
2
10-32 Socket
Head Cap
Screw
McMaster-
Carr
90128A944 Attach baseplate
to endcap
1
Steel Hex
Nut
McMaster-
Carr
90480A195 Attach beam
insert to frame
1
10-32 Split
Washer
McMaster-
Carr
91102A740 Prevent loosening
of components
from vibrations
1
10-32 Washer McMaster-
Carr
92141A011 Load distribution
of beam insert
with respect to
frame
1
M4 8 mm
Alloy Steel
Socket Head
Screw
McMaster-
Carr
91274A115 Attach motor to
enclosure plate
1
M3 8 mm
Alloy Steel
Socket Head
Screw
McMaster-
Carr
91274A103 Attach encoder
to enclosure
1
147
M3 10 mm
Alloy Steel
Socket Head
Screw
McMaster-
Carr
91274A105 Attach enclosure
plate to enclosure
1
ODrive Pro ODrive Store N/A Motor Controller 4
USB Isolator ODrive Store N/A Prevent ground
loops to USB
device
4
ODrive Pro
Heat
Spreader
Plate
ODrive Store N/A Dissipate heat 4
Connector
Pack for
ODrive
ODrive Store N/A Connection wires
from ODrive to
Arduino
4
AMT212B-
V-OD
ODrive Store N/A Absolute encoder 4
Arduino
Giga R1
Arduino
Store
ABX00063 CDCM controller 1
Arduino
Shield
Arduino
Store
A000080 Mounting
headers for
connection to
ODrive
1
Arduino
10 pin
Stackable
Header
Sparkfun
Store
PRT-11376 Connecting
Arduino Shield to
Arduino
4
148
Arduino
8 pin
Stackable
Header
Sparkfun
Store
PRT-9279 Connecting
Arduino Shield to
Arduino
10
Female
Header
10 pin
Sparkfun
Store
PRT-11896 Routing voltage
and ground to
Nano-Fit
Housing
2
Female
2x18 Header
Sparkfun
Store
PRT-16581 Connecting pins
on Arduino
Shield
2
Nano-Fit
Receptacle
Housing
Digikey 1053071204 Housing for
connecting
ODrive cables to
Arduino
4
Nano-Fit
Vertical
Header
Digikey 1053091304 Header for
connecting
ODrive cables to
Arduino
4
Wire Wrap
30 AWG
Digikey R30-5100 Connecting
headers and pins
on the Arduino
Shield
1
Solder Digikey 24-6040-
0061
Soldering headers
and pins on the
Arduino Shield
1
Hook-up wire
(red)
Digikey 3079
RD005
Power connection
between
components
1
149
Hook-up wire
(black)
Digikey 3079
BK005
Power connection
between
components
2
Ring
connector
14-16 AWG
Digikey 0190540110 Connecting
hook-up wire to
power supply
8
Ring
connector
18-22 AWG
Digikey 0190540037 Connecting AC
power cable to
power supply
6
Female bullet
connector
Digikey 0190390016 Connecting
hook-up wire to
motors
12
AC Power
cable
Digikey P010-012 Power cable from
AC outlet to
power supply
1
Power Supply Digikey LRS-600-24 AC/DC power
converter and
power supply for
ODrives
2
SureStep
Regen Clamp
Automation-
Direct
STP-
DRVA-RC-
050A
Regen clamp for
power between
power supply and
ODrive
4
USB-C to
USB-A Cable
Amazon 124287 Connecting to
Arduino or
ODrive
5
Kevlar Line Amazon EK5583 Cables for
CDCM actuation
1
150
25 ft by 1 in
Tape
Measure
Amazon 30-454 Constructing
tape measure
TRAC boom
1
Polycarbon-
ate beam
Grainger 2XPV6 Beam for CDCM
prototype
1
9.5 mm pearl
hard marker
Vicon N/A Tracking pose of
CDCM
1
Table C.1: Parts list for all components required for construction of the third
iteration of the CDCM.